Answer:
P ( P U T ) = 0.6481
Dependent events
P ( P / T ) = (359/756) = 0.4749
Step-by-step explanation:
Given:
- Pink : P
- Teenagers: T
Find:
- P( T u P )
- Are the events T and R dependent, independent, mutually exclusive, or complementary
- If the events are independent, then we can compare P ( P / T ).
Solution:
a)
For first part we will determine the total outcome for T and P:
Total outcomes P or T = 359 + 252 - 121 = 490
(P U T) = (P) + (T) - (P n T)
The total number of possible outcomes are = 756
Hence, P ( P U T ) = 490 / 756 = 0.6481
b)
We are to investigate how the two events are related to one another:
- Check for dependent events:
P ( T n P ) = P( T ) * P ( P / T )
(121 / 756 ) = (252/756) * ( 121 / 252 )
(121 / 756) = (121/756) ....... Hence, events are dependent
- Check for independent events:
P ( T n P ) = P( T ) * P ( P )
(121 / 756 ) = (252/756) * ( 359 / 756 )
(121 / 756) =/ (359/756) ....... Hence, events are not independent
- Check for mutually exclusive events:
P ( T U P ) = P( T ) + P ( P )
(490 / 756 ) = (252/756) + ( 359 / 756 )
(121 / 756) =/ (611/756) ... Hence, events are not mutually exclusive
Hence, the two events are dependent on each other.
c)
If the events are said to be independent then the event:
P ( P / T ) = P (P)
P ( P / T ) = (359/756) = 0.4749
The probability that a randomly selected survey respondent is a Teenager OR selects Pink (Strawberry) as their favorite Starburst color (flavor) can be calculated by adding the individual probabilities of these two events occurring.
The probability of a respondent being a Teenager is 85/756, and the probability of selecting Pink (Strawberry) as their favorite flavor is 359/756. So, the probability of either event happening is (85/756) + (359/756) = 0.4749.
Based on these calculations, the events "a randomly selected person is a Teenager" and "a randomly selected person selects Pink (Strawberry) as their favorite Starburst color (flavor)" are NOT mutually exclusive events because they can both happen.
They are also NOT complementary events because they don't cover all possible outcomes.
Whether they are independent or dependent events can be determined by comparing P(Pink | Teenager) with the probability of Pink (Strawberry) regardless of age group, P(Pink).
If P(Pink | Teenager) is equal to P(Pink), they are independent events; otherwise, they are dependent.
In this case, P(Pink | Teenager) is 114/85, and P(Pink) is 359/756. Comparing these probabilities, we find that P(Pink | Teenager) is not equal to P(Pink), so the events are dependent.
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A recent survey estimated that 19 percent of all people living in a certain region regularly use sunscreen when going outdoors. The margin of error for the estimate was 1 percentage point. Based on the estimate and the margin of error, which is an appropriate conclusion?
Final answer:
The appropriate conclusion from the survey is that the true percentage of all people who regularly use sunscreen in the region is estimated to be between 18% and 20%, considering the margin of error of 1 percentage point.
Explanation:
Based on the estimate and the margin of error, an appropriate conclusion is that it can be said with a certain level of confidence, usually 95%, that the true percentage of all people living in that region who regularly use sunscreen when going outdoors is between 18% and 20%. The margin of error gives a range around the estimate to account for the sampling error, and it implies that the actual value in the population should fall within this range. This helps to understand the precision of the survey result and allows for the natural variability that comes from the fact that not every individual in the region was surveyed.
The variability also depends on the size of the sample, which relates to the confidence interval. For example, smaller samples tend to have larger margins of error, while larger samples tend to lead to smaller margins of error and thus, more precise estimates of the population parameter.
A fair coin is tossed three times. What is the probability that exactly two heads occur, given that a. the first outcome was a tail b. the first two outcomes were heads c. the first two outcomes were tails
Answer:
a) So 25% probability that exactly two heads occur, given that the first outcome was a tail.
b) 50% probability that exactly two heads occur, given that the first two outcomes were heads.
c) 0% probability that exactly two heads occur, given that the first two outcomes were tails.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
We have the following sample space, that is, the possible outcomes:
In which h is heads and t is tails
h - h - h
h - h - t
h - t - h
h - t - t
t - h - h
t - h - t
t - t - h
t - t - t
What is the probability that exactly two heads occur, given that
a. the first outcome was a tail
Four outcomes in which the first outcome was a tail. They are:
t - h - h
t - h - t
t - t - h
t - t - t
In only 1 one them, exactly two heads occur.
1/4 = 0.25
So 25% probability that exactly two heads occur, given that the first outcome was a tail.
b. the first two outcomes were heads
Two possibilities in which the first two outcomes were heads.
h - h - h
h - h - t
In 1 of them, we have exactly two heads.
1/2 = 0.5
So 50% probability that exactly two heads occur, given that the first two outcomes were heads.
c. the first two outcomes were tails
Two possibilities in which the first two outcomes were tails.
t - t - h
t - t - t
In none of them we have exactly 2 heads.
0% probability that exactly two heads occur, given that the first two outcomes were tails.
The probability of getting exactly two heads in three coin tosses is 5/8.
Explanation:To find the probability of exactly two heads occurring in three coin tosses, we need to consider three different scenarios:
If the first outcome was a tail: In this case, we have two remaining tosses and we need both of them to be heads. The probability of getting a head on any single toss is 1/2, so the probability of getting two heads after a tail is (1/2) * (1/2) = 1/4.
If the first two outcomes were heads: In this case, we have one remaining toss and we need it to be a tail. The probability of getting a tail on the remaining toss is 1/2, so the probability of getting exactly two heads after two heads is (1/2) = 1/2.
If the first two outcomes were tails: In this case, we again have one remaining toss and we need it to be a head. The probability of getting a head on the remaining toss is 1/2, so the probability of getting exactly two heads after two tails is (1/2) = 1/2.
Adding up the probabilities of these three scenarios, the overall probability of getting exactly two heads in three coin tosses is 1/4 + 1/2 + 1/2 = 5/8.
The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. Also, note that the vector ai + bj is perpendicular to the line ax + by = c Find the acute angles between the lines: x + √3y = 1 and (1 - √3)x + (1 + √3)y = 8 Thank you!
Answer:
[tex]45^{\circ}[/tex]
Step-by-step explanation:
We are given that two lines equation
[tex]x+\sqrt 3y=1[/tex]...(1)
[tex](1-\sqrt 3)y+(1+\sqrt 3)y=8[/tex]
Compare with the equation of line
ax+by+c=0
[tex]a_1=1,b_1=\sqrt 3[/tex]
[tex]a_2=(1-\sqrt 3),b_2=(1+\sqrt 3)[/tex]
The angle between two lines =Angle between two vectors
The angle between two vector
[tex]a_1i+b_1j[/tex] and
[tex]a_2i+b_2j[/tex]
is given by
[tex]cos\theta=\frac{a_1a_2+b_1b_2}{\sqrt{a^2_1+b^2_1}\sqrt{a^2_2+b^2_2}}[/tex]
Using the formula
Therefore, the angle between two lines
[tex]cos\theta=\frac{1(1-\sqrt 3)+\sqrt 3(1+\sqrt 3)}{\sqrt{(1)+(\sqrt 3)^2}\times \sqrt{(1-\sqrt 3)^2+(1+\sqrt 3)^2}}[/tex]
[tex]cos\theta=\frac{1-\sqrt 3+\sqrt 3+3}{\sqrt{1+3}\times\sqrt{1+3-2\sqrt 3+1+3+2\sqrt 3}}[/tex]
[tex]cos\theta=\frac{4}{2\times\sqrt 8}=\frac{2}{2\sqrt 2}[/tex]
[tex]cos\theta=\frac{1}{\sqrt 2}[/tex]
[tex]cos\theta=cos45^{\circ}[/tex]
By using [tex]cos45^{\circ}=\frac{1}{\sqrt 2}[/tex]
[tex]\theta=45^{\circ}[/tex]
Hence, the angle between two lines =45 degree
Find the probability for the experiment of tossing a six-sided die twice.
1. The sum is 4.
2. The sum is 6.
3. The sum is at least 7.
4. The sum is at least 8.
5. The sum is less than 11.
6. The sum is 2, 3, or 12.
7. The sum is odd and no more than 7.
8. The sum is odd or prime.
Answer:
1. 1/12 or 0.083
2. 5/36 or 0.1389
3. 7/12 or 0.5833
4. 5/12 or 0.4167
5. 11/12 or 0.9167
6. 1/9 or 0.1111
7. 1/3 or 0.3333
8. 5/12 or 0.4167
Step-by-step explanation:
This problem can easily be understood by making sample space first.
Outcomes in sample space=S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
Total outcomes=36
The probability in the following scenario is calculated as
Probability=number of outcomes/Total outcomes
1.
The sum is 4
The sum is 4 ={(1,3),(2,2),(3,1)}
number of outcomes=3
P(The sum is 4)=3/36=1/12 or 0.083
2.
The sum is 6
The sum is 6 ={(1,5),(2,4),(3,3),(4,2),(5,1)}
number of outcomes=5
P(The sum is 6)=5/36 or 0.1389
3.
The sum is at least 7
The sum is at least 7=The sum is greater than or equal to 7
The sum is at least 7={(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,3),(4,4),(4,5),(4,6),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=21
P(The sum is at least 7)=21/36
P(The sum is at least 7)=7/12 or 0.5833
4.
The sum is at least 8
The sum is at least 8=The sum is greater than or equal to 8
The sum is at least 8={(2,6),(3,5),(3,6),(4,4),(4,5),(4,6),(5,3),(5,4),(5,5),(5,6),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=15
P(The sum is at least 8)=15/36
P(The sum is at least 8)=5/12 or 0.4167
5.
The sum is less than 11
The sum is less than 11={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(6,1),(6,2),(6,3),(6,4)}
number of outcomes=33
P(The sum is less than 11)=33/36
P(The sum is less than 11)=11/12 or 0.9167
6.
The sum is 2, 3, or 12
The sum is 2, 3, or 12={(1,1),(1,2),(2,1),(6,6)}
number of outcomes=4
P(The sum is 2, 3, or 12)=4/36
P(The sum is 2, 3, or 12)=1/9 or 0.1111
7.
The sum is odd and no more than 7
The sum is odd and no more than 7={(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(6,1)}
number of outcomes=12
P(The sum is odd and no more than 7)=12/36
P(The sum is odd and no more than 7)=1/3 or 0.3333
8.
The sum is odd or prime
The sum is odd or prime={(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5)}
number of outcomes=15
P(The sum is odd or prime)=15/36
P(The sum is odd or prime)=5/12 or 0.4167
Probabilities are,
1. The sum is 4 = 1/12
2. The sum is 6 = 5/36
3. The sum is at least 7 = 7/12
4. The sum is at least 8 = 5/12
5. The sum is less than 11 = 11/12
6. The sum is 2, 3, or 12 = 1/9
7. The sum is odd and no more than 7 = 1/3
8. The sum is odd or prime = 19/36
When six sided die toss two times, Then total possible Outcomes shown below
[tex](1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\\\(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\\\(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}[/tex]
Total number of outcomes=36
Probability is calculated by dividing number of favourable outcomes by Total number of outcomes
1. Favourable outcomes for the sum is 4 [tex]={(1,3),(2,2),(3,1)}[/tex]
number of outcomes=3
The probability of The sum is 4. = [tex]3/36=1/12 = 0.083[/tex]
2.Favourable outcomes for the sum is 6[tex]={(1,5),(2,4),(3,3),(4,2),(5,1)}[/tex]
number of outcomes=5
The probability of The sum is 6, =5/36
3.The sum is at least 7,
={(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,3),(4,4),(4,5),(4,6),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=21
Probability of the sum is at least 7 =21/36=7/12 = 0.5833
4.The sum is at least 8
The sum is at least 8={(2,6),(3,5),(3,6),(4,4),(4,5),(4,6),(5,3),(5,4),(5,5),(5,6),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=15
The sum is at least 8=15/36 = 5/12 = 0.4167
5.The sum is less than 11
The sum is less than 11={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(6,1),(6,2),(6,3),(6,4)}
number of outcomes=33
The sum is less than 11=33/36 = 11/12
6.The sum is 2, 3, or 12={(1,1),(1,2),(2,1),(6,6)}
number of outcomes=4
The sum is 2, 3, or 12=4/36 = 1/9
7.The sum is odd and no more than 7={(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(6,1)}
number of outcomes=12
The sum is odd and no more than 7=12/36 = 1/3
8.The sum is odd or prime={(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5), (3, 6), (6, 3), (4,5), (5, 4)
number of outcomes=19
The sum is odd or prime=19/36
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A consumer products company found that 44% of successful products also received favorable results from test market research, whereas 13% had unfavorable results but nevertheless were successful. That is, P(successful product and favorable test market) = 0.44 and P(successful product and unfavorable test market) = 0.13. They also found that 32% of unsuccessful products had unfavorable research results, whereas 11% of them had favorable research results, that is P(unsuccessful product and unfavorable test market) = 0.32 and P(unsuccessful product and favorable test market) = 0.11.
Find the probabilities of successful and unsuccessful products given known test market results, that is, P(successful product given favorable test market), P(successful product given unfavorable test market), P(unsuccessful product given favorable test market), and P(unsuccessful product given unfavorable test market).
Answer:
0.800.2890.200.711Step-by-step explanation:
Given:
[tex]P(S\cap F)=0.44\\P(S\cap F^{c})=0.13\\P(S^{c}\cap F^{c}) = 0.32\\P(S^{c}\cap F) = 0.11[/tex]
The rule of total probability states that:
[tex]P(A) = P(A\cap B) + P(A\cap B^{c})[/tex]
Compute the individual probabilities as follows:
[tex]P(S) = P(S\cap F) + P(S\cap F^{c})\\=0.44+0.13\\0.57[/tex]
[tex]P(S^{c}) = 1 - P(S)\\=1-0.57\\=0.43[/tex]
[tex]P(F) = P(S\cap F) + P(S^{c}\cap F)\\=0.44+0.11\\=0.55[/tex]
[tex]P(F^{c})=1-P(F)\\=1-0.55\\=0.45[/tex]
Conditional probability of an event A given B is:
[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}[/tex]
Compute the value of [tex]P(S|F)[/tex]:[tex]P(S|F)=\frac{P(S\cap F)}{P(F)}\\=\frac{0.44}{0.55}\\=0.80[/tex]
Compute the value of [tex]P(S|F^{c})[/tex][tex]P(S|F^{c})=\frac{P(S\cap F^{c})}{P(F^{c})}\\=\frac{0.13}{0.45}\\=0.289[/tex]
Compute the value of [tex]P(S^{c}|F)[/tex][tex]P(S^{c}|F)=\frac{P(S^{c}\cap F)}{P(F}\\=\frac{0.11}{0.55}\\=0.20[/tex]
Compute the value of[tex]P(S^{c}|F^{c})[/tex][tex]P(S^{c}|F^{c})=\frac{P(S^{c}\cap F^{c})}{P(F^{c})}\\=\frac{0.32}{0.45}\\=0.711[/tex]
The probability of successful product given favorable test market is 0.80, and the probability of successful product given unfavorable test market is 0.29. The probability of unsuccessful product given favorable test market is 0.20, and the probability of unsuccessful product given unfavorable test market is 0.71.
Explanation:To solve this, we first need to determine the total probability of each market test outcome. The probability that the market test is favorable (P(favorable test market)) can be found by summing the probabilities of a successful product with a favorable test market and an unsuccessful product with a favorable test market. Therefore, P(favorable test market) = 0.44 + 0.11 = 0.55. Similarly, P(unfavorable test market) = 0.13 + 0.32 = 0.45.
To find the conditional probabilities, we use the formula P(A|B) = P(A and B) / P(B):
P(successful product given favorable test market) = P(successful product and favorable test market) / P(favorable test market) = 0.44 / 0.55 = 0.80.P(successful product given unfavorable test market) = P(successful product and unfavorable test market) / P(unfavorable test market) = 0.13 / 0.45 = 0.29.P(unsuccessful product given favorable test market) = P(unsuccessful product and favorable test market) / P(favorable test market) = 0.11 / 0.55 = 0.20.P(unsuccessful product given unfavorable test market) = P(unsuccessful product and unfavorable test market) / P(unfavorable test market) = 0.32 / 0.45 = 0.71.Learn more about Conditional Probability here:https://brainly.com/question/32171649
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If a student determined that the volume of a sample of water is 11.4 mL, and its mass is 10.98 g, what should the student record as the experimental density of the water sample, with the correct number of significant figures
Answer:1.04 ml/g.
Step-by-step explanation:
The formula we use to find the density :
[tex]\text{Density}=\dfrac{\text{Volume}}{\text{Mass}}[/tex]
We are given that ,
Volume of a sample of water = 11.4 mL
Mass of sample of water = 10.98 g
Then , the experimental density of the water sample will be :-
[tex]\text{Density}=\dfrac{11.4}{10.98}=1.03825136\approx1.04[/tex]
Hence, the experimental density of the water sample is 1.04 ml/g.
Following are measurements of soil concentrations (in mg/kg) of chromium (Cr) and nickel (Ni) at 20 sites in the area of Cleveland, Ohio. These data are taken from the article "Variation in North American Regulatory Guidance for Heavy Metal Surface Soil Contamination at Commercial and Industrial Sites" (A. Jennings and J. Ma, J Environment Eng, 2007:587–609).
Cr: 34 1 511 2 574 496 322 424 269 140 244 252 76 108 24 38 18 34 3O 191
Ni: 23 22 55 39 283 34 159 37 61 34 163 140 32 23 54 837 64 354 376 471
(a) Construct a histogram for each set of concentrations.
(b) Construct comparative boxplots for the two sets of concentrations.
(c) Using the boxplots, what differences can be seen between the two sets of concentrations?
Answer:
a) see attached
b) see attached
Step-by-step explanation:
There is existence of outlier in the Ni data and there is none is Cr data.
########################################
# You can try this out in R programming
cr = c(34, 1, 511, 2, 574, 496, 322, 424, 269, 140, 244, 252, 76, 108, 24,
38, 18, 34, 30, 191)
Ni = c(23, 22, 55, 39, 283, 34, 159, 37, 61, 34, 163, 140, 32,
23, 54, 837, 64, 354, 376, 471)
par(mfrow=c(1,2))
hist(cr, col='green')
hist(Ni, col='brown')
par(mfrow=c(1,2))
boxplot(cr, main = 'Boxplot of Cr')
boxplot(Ni, main = 'Boxplot of Ni')
boxplot(cr, Ni)
The question involves constructing histograms and boxplots for two sets of data on soil concentrations of certain metals. After construction, these should be analysed for differences in central tendency, spread, variability, skewness, or presence of outliers.
Explanation:This question seems to be related to Mathematics, specifically Statistics and Data Analysis. Firstly, to construct a histogram, you need to group the data into 'bins' or 'intervals', count how many data points fall into each bin, and then plot these counts as bars in your histogram. It's similar for both Chromium (Cr) and Nickel (Ni) data.
To construct comparative boxplots, it's essential to: calculate the quartiles (Q1, Q2 aka median, and Q3), identify the minimum and maximum points, and define possible outliers. With these statistical measures, you can draw the boxplots for both sets and analyse them. The dataset Cr seems to have several values clustered in the low range while Ni seems to have more spread, which can be verified by the boxplot comparison.
The interpretation of the boxplots should reflect on any differences in central tendency and spread between the two sets of concentrations. The variability, skewness, or outliers could also be significantly different. Any such inferred details from the boxplots may hint towards different contamination levels or heterogeneity in each of the datasets.
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The brain volumes (cm cubed) of 20 brains have a mean of 1105.1 cm cubed and a standard deviation of 123.5 cm cubed. Use the given standard deviation and the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of 1332.1 cm cubed be significantly high? Significantly low values are 858 cm cubed or lower. (Type an integer or a decimal. Do not round.)
Answer:
[tex]1332.1 \:cm^3[/tex] is neither significantly low nor significantly high.
Step-by-step explanation:
From the information given, we know that:
The mean is [tex]\bar{x}=1105.1 \:cm^{3}[/tex] and the standard deviation is [tex]\sigma=123.5 \:cm^{3}[/tex].
The range rule of thumb says that:
[tex]minimum \:usual \:value=\bar{x}-2\sigma\\\\maximum \:usual \:value=\bar{x}+2\sigma[/tex]
Applying these definitions, we get that
[tex]minimum \:usual \:value=1105.1-2(123.5)=858.1 \:cm^3\\\\maximum \:usual \:value=1105.1+2(123.5)=1352.1 \:cm^3[/tex]
We note that [tex]1332.1 \:cm^3[/tex] is between [tex]858.1 \:cm^3[/tex] and [tex]1352.1 \:cm^3[/tex], which indicates [tex]1332.1 \:cm^3[/tex] is neither significantly low nor significantly high.
PLEASE HELP!!!! ill give 100 points!!
SHOW ALL WORK PLEASE AND THANK YOU
The area of a triangle is 1/2 x base x height
Area = 1/2 x 24 x 28 = 336 square inches.
Because the corners are rounded the are would actually be just under 336 but no radius was given to get the exact area.
Answer:
yes
Step-by-step explanation:
Suppose x is a normally distributed random variable with µ = 56 and σ = 3. Find a value x0 of the random variable x that satisfies the following equations or statements.a. 10% of the values of x are less than x0.b. 80% of the values of x are less than x0.c.1% of the values of x are greater than x0.
Answer:
a) 52.15
b) 58.53
c) 62.98
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 56
Standard Deviation, σ = 3
We are given that the distribution of x is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) We have to find [tex]x_0[/tex] such that
P(X < x) = 0.1
[tex]P( X < x) = P( z < \displaystyle\frac{x_0 - 56}{3})=0.1[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < -1.282) = 0.1[/tex]
[tex]\displaystyle\frac{x_0 - 56}{3} = -1.282\\\\x_0 = 52.15[/tex]
b) We have to find [tex]x_0[/tex] such that
P(X < x) = 0.8
[tex]P( X < x) = P( z < \displaystyle\frac{x_0 - 56}{3})=0.8[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < 0.842) = 0.8[/tex]
[tex]\displaystyle\frac{x_0 - 56}{3} = 0.842\\\\x_0 = 58.53[/tex]
c) We have to find [tex]x_0[/tex] such that
P(X > x) = 0.01
[tex]P( X > x) = P( z > \displaystyle\frac{x - 56}{3})=0.01[/tex]
[tex]= 1 -P( z \leq \displaystyle\frac{x - 56}{3})=0.01 [/tex]
[tex]=P( z \leq \displaystyle\frac{x - 56}{3})=0.99 [/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < 2.326) = 0.99[/tex]
[tex]\displaystyle\frac{x_0 - 56}{3} = 2.326\\\\x_0 = 62.98[/tex]
To find specific percentiles for a normally distributed variable x with mean 56 and standard deviation 3, you must use a z-score table or calculator to find the corresponding z-scores for the 10%, 80%, and top 1% percentages, then use the formula x0 = µ + z*σ to calculate x0.
Explanation:To find the value x0 of the normally distributed random variable x with mean µ = 56 and standard deviation σ = 3, that corresponds to specific percentiles, one would typically use a z-score table (or a calculator with a normal distribution function). However, in this scenario, the values provided for the x variable and the associated z-scores in the question are incorrect and would not provide a reasonable answer. Instead, I will explain the steps needed to find x0 without using the incorrect values provided.
Finding x0 for Each Given Percentile:For the 10% percentile: Find the z-score that corresponds to a cumulative probability of 0.10.After finding each z-score, use the formula x0 = µ + z*σ to calculate the corresponding x0 for each case.
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Subtracting fractions with unlike denominators 10/12 -1/4 show work
Answer:
7/12
Step-by-step explanation:
Step 1: Find common denominators
The common denominator is 12
10/12 is already good
1/4 * 3/3 = 3/12
Step 2: Subtract
10/12 - 3/12 = 7/12
Answer: 7/12
Final answer:
To subtract the fractions 10/12 and 1/4, find a common denominator, convert the fractions, and then subtract the numerators, resulting in the answer 7/12.
Explanation:
To subtract fractions with unlike denominators, you first need to find a common denominator. For the fractions 10/12 and 1/4, the common denominator is 12. You can convert 1/4 to 3/12, because 4 times 3 equals 12. Now you can subtract the fractions as follows:
10/12 - 3/12 = (10 - 3)/12 = 7/12
So, the answer to 10/12 - 1/4 is 7/12.
This process relies on the intuitive understanding that you can only subtract numerators if the denominators are the same. We never add or subtract the denominators themselves. When your fractions share the same denominator, the subtraction is straightforward as we are simply dealing with parts of the whole that are of the same size.
How do you convert $150,000 US dollars to $240,000 Australian dollars (1.31545 AU) using the exchange rate per US dollar?
Answer:
240,000 AUD to USD = 161,924.79 US Dollars or 150,000 USD to AUD = 222,316.63 Australian Dollars
Step-by-step explanation:
With a full tank of gas, if you can drive 485 miles, your car uses 18 miles per gallon. Write an equation to model this situation (use m for miles you can drive and g for gallons in tank).
Answer: m= 18g
Step-by-step explanation:
Let m= miles you can drive
g = g for gallons in tank
We know that there is directly proportional relation between the distance traveled by vehicle and the number of gallons of gas.
Then, rate of miles driven by you per gallon = [tex]\dfrac{m}{g}[/tex]
Since , rate of miles driven by you per gallon = 18 miles per gallon (given)
Then,
[tex]\dfrac{m}{g}=18\\\\ m= 18g[/tex]
Hence, the equation to model this situation : m= 18g
A newly proposed endoscopy method used to screen for colon cancer is tested among a random sample of the population. Given that 16 individuals from a test population of 100,000 individuals has colon cancer and 14 individuals test positive for colon cancer, the researchers should determine the sensitivity of the newly proposed screening method is 0.875.
Please explain why this is true
Answer:
The value of sensitivity obtained is equal to the one given in statement.
Step-by-step explanation:
Sensitivity is the ratio of number of positive test results to the number of individuals having disease. It is the ability of the test to detect all those with disease in the screened population.
Its formula is written as:
Sensitivity = [tex]\frac{number of true positives}{total with disease}[/tex]
So, in this case we have values
sensitivity = 14/16 = 0.875 So, the given statement is true.
You select a sample of 50 scores from a population of 2,000 scores. You compute the range and standard deviation on the sample of 50 scores. You then select another sample of 50 scores from the same population. What measure of dispersion is likely to vary most between your first and second samples?
Answer:
The measure of dispersion which is likely to vary most between your first and second samples is the range.
Step-by-step explanation:
The range and standard deviation of a data are measures of dispersion, i.e. they measure the degree to which the data is dispersed.
The formula to compute the range is:
[tex]Range=X_{max}-X_{min}[/tex]
The formula to compute the sample standard deviation is:
[tex]s=\sqrt{\frac{1}{n-1}\sum (X-\bar X)^{2} }[/tex]
The sample size is: n = 50.
As the sample size is large (n = 50 > 30) the sample standard deviation (s) can be used to approximate the population standard deviation (σ). Thus, whatever the sample values be both the standard deviations can be used to approximate the population standard deviation. Hence, it can be said that both the sample standard deviations are approximately equal.Whereas the range of the two samples are very likely to vary since it is based on the minimum and maximum value of the data. For both the samples the minimum and maximum value may be differ. Thus providing different range values.Thus, the measure of dispersion which is likely to vary most between your first and second samples is the range.
Final answer:
The range is more likely to vary between two samples from a population due to its sensitivity to extreme values, whereas the standard deviation is a more stable measure of dispersion that accounts for all values in the dataset.
Explanation:
The measure of dispersion most likely to vary between the first and second samples from the population is the range. The range is sensitive to the particular values in the dataset because it only takes into account the smallest and largest values. Since each sample may have a different set of extreme values, the range can vary greatly from sample to sample. In contrast, the standard deviation is less likely to vary significantly between samples, as it measures the amount of variation or dispersion from the mean of a dataset and takes into account all the values in the set.
When drawing multiple samples from a population, the distribution of the sample means will tend to form a normal distribution around the population mean according to the Central Limit Theorem. In the case of a normally distributed population with a known mean and standard deviation, the sampling distribution of the sample mean will have a standard deviation equal to the population standard deviation divided by the square root of the sample size (known as the standard error).
Suppose x is a random variable best described by a uniform probability distribution with c=20 and d=40.
Find the probability P(20≤x≤35).
Answer:
P(20≤x≤35) = 0.75 .
Step-by-step explanation:
We know that the probability distribution function of Uniform Distribution is represented as :
If x follows Uniform(c,d) then,
f(x) = [tex]\frac{1}{d-c}[/tex] where c < x < d
To fond the given probability it is better to first calculate the Cumulative Distribution Function(CDF) of Uniform Distribution.
The CDF of Uniform Distribution is P(X<=x) = [tex]\frac{x-c}{d-c}[/tex] where d > c .
Therefore, P(20<=x<=35) = P(x<=35) - P(x<20)
P(x<=35) = [tex]\frac{x-20}{40-20}[/tex] because we are given c = 20 and d = 40.
= [tex]\frac{35-20}{40-20}[/tex] = 0.75
P(x<20) = [tex]\frac{x-20}{40-20}[/tex] = [tex]\frac{20-20}{40-20}[/tex] = 0
Hence, P(20<=x<=35) = 0.75 - 0 = 0.75.
The probability of x being between 20 and 35 is 0.75 or 75%.
Explanation:The uniform probability distribution is defined as:
P(x) = 1/(b-a) for a ≤ x ≤ b and 0 otherwise.
In this case, a=20 and b=40, so the probability P(20≤x≤35) can be calculated as:
P(20≤x≤35) = (35-20)/(40-20) = 15/20 = 0.75.
Therefore, the probability of x being between 20 and 35 is 0.75 or 75%.
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At a used dealership, let X be an independent variable representing the age in years of a motorcycle and Y be the dependent variable representing the selling price of used motorcycle. The data is now given to you.
X = {5, 10; 12, 14, 15}; Y = {500, 400, 300, 200, 100}
(a) Write the regression model.
(b) Estimate the parameters of the model
(c) Write the prediction equation
(d) Calculate SSE.
Answer:
a) Selling price y= a + b (age x)
b)
a= 728.025
b= -38.217
c)
Selling price y = 728.025 - 38.217 age x
d)
SSE=8280.25
Step-by-step explanation:
a)
The regression model can be written as
y=a+bx
Here y=selling price and x is age.
So, the regression model will be
Selling price y= a + b (age x)
b)
We have to find the values of "a" and "b"
[tex]b=\frac{nsumxy-(sumx)(sumy)}{nsumx^{2} -(sumx)^2}[/tex]
sumx=5+10+12+14+15=56
sumy=500+400+300+200+100=1500
sumxy=5*500+10*400+12*300+14*200+15*100=14400
sumx²=5²+10²+12²+14²+15²=690
n=5
[tex]b=\frac{5(14400)-(56)(1500)}{5(690) -(56)^2}[/tex]
b=-12000/314
b=-38.217
ybar=a+bxbar
a=ybar-bxbar
ybar=sumy/n=1500/5=300
xbar=sumx/n=56/5=11.2
a=300-(-38.217)(11.2)
a=300+428.025
a=728.025
c)
Selling price y = a - b(age x)
Selling price y = 728.025 - 38.217 age x
d)
SSE known as sum of square of error can be calculated as
SSE=sum(y-yhat)²
y 500 400 300 200 100
x 5 10 12 14 15
yhat= 728.025 - 38.217 age x 536.940 345.855 269.421 192.987 154.770
y-yhat -36.940 54.145 30.579 7.013 -54.770
(y-yhat)² 1364.56 2931.68 935.08 49.18 2999.75
SSE=sum(y-yhat)²
SSE=1364.56 +2931.68 +935.08 +49.18 +2999.75
SSE =8280.25
A glass of orange juice contains 3.2 dL of juice. How many milliliters of orange juice are in the glass?
Answer:
There are 320 milliliters of orange juice in the glass.
Step-by-step explanation:
Unit conversion problems can be solved by rules of three.
We have that each dL is worth 100 mL(milliliters).
How many milliliters of orange juice are in the glass?
How many ml are 3.2dL of juice?
1 dL - 100 mL
3.2 dL - x mL
[tex]x = 100*3.2[/tex]
[tex]x = 320[/tex]
There are 320 milliliters of orange juice in the glass.
The measure of juice is 320 milliliters.
Given that
A glass of orange juice contains 3.2 dL of juice.
We have to determine
How many milliliters of orange juice is in the glass?
According to the question
To determine the measure of juice in milliliters following all the steps given below.
A glass of orange juice contains 3.2 dL of juice.
Here 1 dL = 100 milliliters
Then,
The measure of juice in milliliters is,
[tex]\rm = 3.2 \times 100\\\\= 320 \ milliliters[/tex]
Hence, The measure of juice is 320 milliliters.
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In the field of quality control, the science of statistics is often used to determine if a process is "out of control." Suppose the process is, indeed, out of control and 15% of items produced are defective. (a) If two items arrive off the process line in succession, what is the probability that both are defective? (b) If three items arrive in succession, what is the probability that two are defective?
Answer:
a) P=0.0225
b) P=0.057375
Step-by-step explanation:
From exercise we have that 15% of items produced are defective, we conclude that probabiity:
P=15/100
P=0.15
a) We calculate the probabiity that two items are defective:
P= 0.15 · 0.15
P=0.0225
b) We calculate the probabiity that two of three items are defective:
P={3}_C_{2} · 0.85 · 0.15 · 0.15
P=\frac{3!}{2!(3-2)!} · 0.019125
P=3 · 0.019125
P=0.057375
Using the binomial distribution, it is found that there is a:
a) 0.0225 = 2.25% probability that both are defective.
b) 0.0574 = 5.74% probability that two are defective.
For each item, there are only two possible outcomes, either it is defective, or it is not. The probability of an item being defective is independent of any other item, hence, the binomial distribution is used to solve this question.
Binomial probability distribution
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
x is the number of successes. n is the number of trials. p is the probability of a success on a single trial.In this problem, 15% of items produced are defective, hence [tex]p = 0.15[/tex].
Item a:
Two items, hence [tex]n = 2[/tex].
The probability is P(X = 2), then:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 2) = C_{2,2}.(0.15)^{2}.(0.85)^{0} = 0.0225[/tex]
0.0225 = 2.25% probability that both are defective.
Item b:
Three items, hence [tex]n = 3[/tex].
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 2) = C_{3,2}.(0.15)^{2}.(0.85)^{1} = 0.0574[/tex]
0.0574 = 5.74% probability that two are defective.
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PLZ HELP A delivery truck is transporting boxes of two sizes: large and small. The large boxes weigh 55 pounds each, and the small boxes weigh 35 pounds each. There are 125 boxes in all. If the truck is carrying a total of 4950 pounds in boxes, how many of each type of box is it carrying?
Answer:
There were 28.75 large boxes and 96.25 small boxes.
Step-by-step explanation:
Create a system of equations to solve.
State your variables
let x be the number of small boxes
let y be the number of large boxes
35x + 55y = 4950 Equation for weight
x + y = 125 Equation for number of boxes
Rearrange equation for number of boxes to isolate "x".
x = 125 - y
Substitute the new equation into the equation for weight
35x + 55y = 4950
35(125 - y) + 55y = 4950
Expand the brackets
4375 - 35y + 55y = 4950 Combine like terms
4375 + 20y = 4950
Start isolating "y"
4375 + 20y = 4950
4375 - 4375 + 20y = 4950 - 4375 subtract 4375 from both sides
20y = 575
20y/20 = 575/20 divide both sides by 20
y = 28.75 number of large boxes
Calculate "x". Rearrange the equation for number of boxes to isolate "y".
y = 125 - x Substitute this expression
35x + 55y = 4950 equation for weight
35x + 55(125 - x) = 4950 expand the brackets
35x + 6875 - 55x = 4950 combine like terms
6875 - 20x = 4950 start isolating "x"
6875 - 6875 - 20x = 4950 - 6875 subtract 6875 on both sides
-20x = -1925
-20x/-20 = -1925/-20 divide both sides by -20
x = 96.25 number of small boxes
There is a mistake with this question because you should not be able to have partial boxes. However, the answer is:
There were 28.75 large boxes and 96.25 small boxes.
You have decided that you want to be a millionaire when you retire in 44 years. If you can earn an annual return of 11.06 percent, how much do you have to invest today? What if you can earn an annual return of 5.53 percent?
Answer:
1. Let amount deposited now be P.
So P*(1.1106^44) = 1,000,000, which means P=9896.06
2. Let amount deposited now be P.
So P*(1.0553^44) = 1,000,000, which means P=93639.42
Step-by-step explanation:
Answer:
Step-by-step explanation:
Let P represent the amount that you need to invest today. Thus, principal = $P
The return would be compounded annually. So
n = 1
The rate at which the principal would be compounded is 11.06%. So
r = 11.06/100 = 0.1106
The investment would be for 44 years. So
t = 44
The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of t years.
A = $1000000
Therefore
1000000 = P(1+0.1106/1)^1×44
1000000 = P(1.1106)^44
P = 1000000/101.05
P = $9896.1
2) if you can earn an annual return of 5.53 percent, then
r = 5.53/100 = 0.053
1000000 = P(1+0.53/1)^1×44
1000000 = P(1.053)^44
P = 1000000/9.7
P = $103093
In Saras class, 2/5 of the students ride a bus, and 1/3 ride a car to school. The rest walk to school. Explain how you can find the fraction of students who walk to school. Find the fraction of students who ride a bus or car to school. Draw a diagram and use an equation to find your answer. Find the fraction of students who walk to scool. Draw a diagram and use an equation to find your answer.
Answer:
See explanation
Step-by-step explanation:
In Saras class, [tex]\frac{2}{5}[/tex] of the students ride a bus, and [tex]\frac{1}{3}[/tex] ride a car to school.
The rest
[tex]1-\dfrac{2}{5}-\dfrac{1}{3}=\dfrac{15-6-5}{15}=\dfrac{4}{15}[/tex]
walk to school (simply subtract from one whole the fractions of students).
The fraction of students who ride a bus or car to school is
[tex]\dfrac{2}{5}+\dfrac{1}{3}=\dfrac{6+5}{15}=\dfrac{11}{15}.[/tex]
See attached diagram for pictorial representations (red diagram shows the fraction of students who ride a bus, blue - ride a car, green - walk to school)
The fraction of students who walk to school is [tex]$\boxed{\frac{4}{15}}$[/tex].
To find the fraction of students who walk to school, we first consider the fractions of students who ride a bus or car to school. According to the information given:
- The fraction of students who ride a bus is [tex]$\frac{2}{5}$[/tex].
- The fraction of students who ride a car is [tex]$\frac{1}{3}$[/tex].
To find the total fraction of students who use transportation (bus or car), we add these two fractions together. However, we must first find a common denominator to add them properly. The least common multiple (LCM) of 5 and 3 is 15, so we convert both fractions to have a denominator of 15:
[tex]- $\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$ (for the bus riders)[/tex]
[tex]- $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$ (for the car riders)[/tex]
Now we can add these fractions:
[tex]- $\frac{6}{15} + \frac{5}{15} = \frac{11}{15}$[/tex]
This represents the fraction of students who either ride a bus or a car to school. To find the fraction of students who walk to school, we subtract this fraction from the whole, which is 1 (since the whole class is represented by 1, or [tex]$\frac{15}{15}$):[/tex]
[tex]- $1 - \frac{11}{15} = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}$[/tex]
Therefore, the fraction of students who walk to school is [tex]$\frac{4}{15}$[/tex].
To visualize this with a diagram, we can imagine a whole circle representing the entire class. We divide this circle into 15 equal parts (since our denominators are 15). We shade 6 parts to represent the bus riders and another 5 parts to represent the car riders. The remaining unshaded parts, which are 4 in number, represent the students who walk to school.
The equation representing the total fraction of students who walk or use transportation is:
[tex]\[ \text{Bus riders} + \text{Car riders} + \text{Walkers} = \text{Whole class} \] \[ \frac{6}{15} + \frac{5}{15} + \text{Walkers} = 1 \][/tex]
Solving for Walkers:
[tex]\[ \text{Walkers} = 1 - \left( \frac{6}{15} + \frac{5}{15} \right) \] \[ \text{Walkers} = 1 - \frac{11}{15} \] \[ \text{Walkers} = \frac{15}{15} - \frac{11}{15} \] \[ \text{Walkers} = \frac{4}{15} \][/tex]
Thus, the fraction of students who walk to school is[tex]$\boxed{\frac{4}{15}}$.[/tex]
What is the simplified form of this expression?
(-3x2 + 4x) + (2x2 − x − 11)
Answer:
option b
-x^2 + 3x - 11
Step-by-step explanation:
we have to add the X of the same order
x^2
-3x^2 + 2x^2 = -x^2
x^1
4x + ( -x ) = 3x
x^0
0 + (-11) = -11
finally we accommodate and get the final result
-x^2 + 3x - 11
Answer:-x^2 + 3x - 11
Step-by-step explanation:
I got it right on the test
A study of 31,000 hospital admissions in New York State found that 4% of the admissions led to treatment-caused injuries. One-seventh of these treatment-caused injuries resulted in death, and one-fourth were caused by negligence. Malpractice claims were filed in one out of 7.5 cases involving negligence, and payments were made in one out of every two claims.a. What is the probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence? b. What is the probability a person admitted to the hospital will die from a treatment-caused injury?c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid?
Answer:
Step-by-step explanation:
a) probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence
P(injury) = 4%
P(negligence) = 1/4 = 0.25
We need to find probability (injury)(negligence)
P(injury) * P(negligence) = 0.04*0.25 = 0.01
b) probability a person admitted to the hospital will die from a treatment-caused injury
P(injury) = 4%
P(death) = 1/7
P(Injury) *P(death) = 0.04/7 = 0.00571
c) In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid
P(claim) = 1/7.5
P(payment) = 1/2
P(claim)*P(payment) = 1/7.5 * 1/2 = 0.06
Answer:
a.) 0.01
b.) 0.006
c.)0.00067
Step-by-step explanation:
Suppose that we have a standard Solow model with a Cobb-Douglas production function. The central equation of the model is as follows: kt+1 = sAkα t + (1 − δ)kt . Consumption per worker is given by: ct = (1 − s)Akα t . (a) Solve for an expression for the steady state capital stock per worker. In doing so, assume that the level of productivity is fixed at some value A.
Answer:
The answer is shown in the attachment
Step-by-step explanation:
The detailed step by step and appropriate mathematical manipulation for the expression for the steady state capital stock per worker is as shown in the attachment.
Find the angle between the vectors. Use a calculator if necessary. (Enter your answer in radians. Round your answer to three decimal places.) (√3,1) and <0, 5 > Find the angle between the vectors. Use a calculator if necessary.<0,4,4> and <3,-3,0>
Answer:
a. The angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(\sqrt{3}, 1\right)[/tex] and [tex]\mathbf{v}=\left(0, 5\right)[/tex] is [tex]\phi=\frac{\pi}{3}[/tex] or [tex]60\º[/tex].
b. The angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(0, 4, 4\right)[/tex] and [tex]\mathbf{v}=\left(3, -3, 0\right)[/tex] is [tex]\phi=\frac{2 \pi}{3}[/tex] or [tex]120\º[/tex].
Step-by-step explanation:
a. To calculate the angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(\sqrt{3}, 1\right)[/tex] and [tex]\mathbf{v}=\left(0, 5\right)[/tex] you must:
Step 1: Calculate the dot product.
The dot product is given as [tex]\displaystyle{\large{{\left({u}_{{x}},{u}_{{y}}\right)}\cdot{\left({v}_{{x}},{v}_{{y}}\right)}={u}_{{x}}\cdot{v}_{{x}}+{u}_{{y}}\cdot{v}_{{y}}}}[/tex].
So,
[tex]\left(\sqrt{3}, 1\right)\cdot\left(0, 5\right)=\left(\sqrt{3}\right)\cdot\left(0\right)+\left(1\right)\cdot\left(5\right)=5[/tex]
Step 2: Find the lengths of the vectors.
[tex]\left|\mathbf{u}\right|=\sqrt{\left(u_x\right)^2+\left(u_y\right)^2}=\sqrt{\left(\sqrt{3}\right)^2+\left(1\right)^2}=2[/tex]
[tex]\left|\mathbf{v}\right|=\sqrt{\left(v_x\right)^2+\left(v_y\right)^2}=\sqrt{\left(0\right)^2+\left(5\right)^2}=5[/tex]
Step 3: The angle is given by [tex]\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}[/tex]
[tex]\frac{5}{2 \cdot 5}=\frac{1}{2}[/tex]
[tex]\phi=cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}=60^0[/tex]
b. To calculate the angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(0, 4, 4\right)[/tex] and [tex]\mathbf{v}=\left(3, -3, 0\right)[/tex] you must:
Step 1: Calculate the dot product.
[tex]\left(0, 4, 4\right)\cdot\left(3, -3, 0\right)=\left(0\right)\cdot\left(3\right)+\left(4\right)\cdot\left(-3\right)+\left(4\right)\cdot\left(0\right)=-12[/tex]
Step 2: Find the lengths of the vectors.
[tex]\left|\mathbf{u}\right|=\sqrt{\left(u_x\right)^2+\left(u_y\right)^2+\left(u_z\right)^2}=\sqrt{\left(0\right)^2+\left(4\right)^2+\left(4\right)^2}=4 \sqrt{2}[/tex]
[tex]\left|\mathbf{v}\right|=\sqrt{\left(v_x\right)^2+\left(v_y\right)^2+\left(v_z\right)^2}=\sqrt{\left(3\right)^2+\left(-3\right)^2+\left(0\right)^2}=3 \sqrt{2}[/tex]
Step 3: The angle is given by [tex]\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}[/tex]
[tex]\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}=\frac{-12}{4 \sqrt{2} \cdot 3 \sqrt{2}}=- \frac{1}{2}[/tex]
[tex]\phi=cos^{-1}\left(- \frac{1}{2}\right)=\frac{2 \pi}{3}=120^0[/tex]
To find the angle between two vectors, use the dot product formula. For the vectors (√3,1) and <0, 5 >, the angle is π/3 radians. For the vectors <0,4,4> and <3,-3,0>, the angle is 2π/3 radians.
Explanation:To find the angle between two vectors, we can use the dot product formula. Let's start with the first set of vectors (√3,1) and <0, 5>. We can calculate the dot product by multiplying the corresponding components and adding them up: (√3)(0) + (1)(5) = 5. Next, we need to calculate the magnitudes of each vector. The magnitude of (√3,1) is √(√3)^2 + 1^2 = √4 = 2. The magnitude of <0, 5> is √0^2 + 5^2 = √25 = 5. Now, we can calculate the angle between the vectors using the formula: cosθ = dot product / (magnitude 1 * magnitude 2). Plugging in the values, cosθ = 5 / (2 * 5) = 1/2. Taking the inverse cosine of 1/2 gives us the angle in radians: θ = acos(1/2) = π/3 ≈ 1.047.
For the second set of vectors <0,4,4> and <3,-3,0>, we will follow the same steps. The dot product is (0)(3) + (4)(-3) + (4)(0) = -12. The magnitude of <0,4,4> is √0^2 + 4^2 + 4^2 = √32 = 4√2. The magnitude of <3,-3,0> is √3^2 + (-3)^2 + 0^2 = √18 = 3√2. Plugging in the values, cosθ = -12 / (4√2 * 3√2) = -1/2. Taking the inverse cosine of -1/2 gives us the angle in radians: θ = acos(-1/2) = 2π/3 ≈ 2.094.
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Determine whether the lines
L1: x=23+6t, y=12+3t, z=19+5t
and
L2: x=-9+7t, y=-7+5t, z=-12+8t
intersect, are skew, or are parallel. If they intersect, determine the point of intersection; if not leave the remaining answer blanks empty.
Do/are the lines:
Point of intersection: (, ,)
Answer:
skew lines
Step-by-step explanation:
Given are two lines in 3 dimension as
[tex]L1: x=23+6t, y=12+3t, z=19+5t\\L2: x=-9+7t, y=-7+5t, z=-12+8t[/tex]
To find out whether parallel or skew or intersect
If parallel direction ratios should be proportional
Direction ratios of I line are 6,3,5 and not proportional to that of II line (7,5,8)
So not parallel
If intersect we must have same point for the two lines
Let us change parameter for II line to s to avoid confusion. If intersecting, then
[tex]23+6t = -9+7s\\12+3t = -7+5s\\19+5t =12+8s\\[/tex]
6t-7s=-32 and 3t-5s = -19
Solving
t=-3 and s =2
Check this with III equation
Left side = 19-15 =4 and right side = 12+16 =28
not equal
So there cannot be any point of intersection.
These two are skew lines
A container initially containing10 L of water in which there is 20 g of salt dissolved. A solution containing 4 g/L of salt is pumped into the container at a rate of 2 L/min, and the well-stilled mixture runs out at a rate of 1 L/min. How much salt is in the tank after 40 min
Answer:
[tex] A(40)= \frac{-200}{10+40} +4 (10 +40)=-4+200 = 196 [/tex]
Step-by-step explanation:
For this case the solution flows at a rate of 2L/min and leaves at 1L/min. So then we can conclude the volume is given by [tex] V= 10 +t[/tex]
Since the initial volume is 10 L and the volume increase at a rate of 1L/min.
For this case we can define A as the concentration for the salt in the container. And for this case we can set up the following differential equation:
[tex] \frac{dA}{dt}= 4 \frac{gr}{L} *2 \frac{L}{min} - \frac{A}{10+t}[/tex]
Because at the begin we have a concentration of 8 gr/L and would be decreasing at a rate of [tex] \frac{A}{10+t}[/tex]
So then we can reorder the differential equation like this:
[tex] \frac{dA}{dt} +\frac{A}{10+t} =8[/tex]
We find the solution using the integration factor:
[tex] \mu = -\int \frac{1}{10+t} dt = -ln(10+t)[/tex]
And then the solution would be given by:
[tex] A = e^{-ln (10+t)} (\int e^{\int \frac{1}{10+t} dt})[/tex]
And if we simplify this we got:
[tex] A= \frac{1}{10+t} (c + \int (10 +t) 8 dt)[/tex]
And after do the integral we got:
[tex] A= \frac{c}{10+t} +4 (10 +t) [/tex]
And using the initial condition t=0 A= 20 we have this:
[tex] 20 = \frac{c}{10} +40[/tex]
[tex] c= -200[/tex]
So then we have this function for the solution of A:
[tex] A= \frac{-200}{10+t} +4 (10 +t) [/tex]
And now replacinf t= 40 we got:
[tex] A(40)= \frac{-200}{10+40} +4 (10 +40)=-4+200 = 196 [/tex]
The amount of salt in the tank after 40 minutes is determined by first calculating the total amount of salt added to the tank, then considering the amount of salt-water mixture that exited the tank during this period due to the outgoing flow.
Explanation:This question relates to a typical problem within the field of differential equations. You are asked to find the amount of salt in the tank after 40 minutes. The initial amount of salt in the tank is 20g, and additional salt-water solution is being pumped in at a rate of 2L/min with a salt concentration of 4g/L, hence adding 8g of salt per minute. At the same time, the well-stirred mixture is flowing out at a rate of 1L/min.
Therefore, after 40 minutes:
Additional salt added = 8g/min * 40 min = 320g.
Total salt content now = initial salt content + added salt = 20g + 320g = 340g.
However, within these 40 minutes,40 L of the mixture has flown out. Since the mixture is well-stirred, it retains the same concentration of salt as the mixture in the tank. Hence, if x is the amount of salt left in the tank, it will be on average the same concentration as that which flowed out and can be represented as follows: x/50 = (340-x)/40.
Solving for x we get the amount of salt left in the tank after 40 minutes.
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I need a better explanation how to find length and area of a triangle
Answer:
Height(h) = 24in
Area = 204in²
Step-by-step explanation:
we would use the heron formula because we have all the sides to calulate the area
S = (A+B+C)/2 = (25 + 17 + 26)/2 = 68/2 = 34
area = √s(s-a)(s-b)(s-c) = √34(34-25)(34-17)(34-26) = √34(9)(17)(8) = √34x9x17x8 = √41616 = 204
area= (1/2)bh
204 = (1/2)bh b= 17, h =?
204 = (1/2)17h
204 x 2 = 17h
408 = 17h
h = 408/17 = 24in
Height(h) = 24in
Area = 204in²
Angela took a general aptitude test and scored in the 91st percentile for aptitude in accounting. (a) What percentage of the scores were at or below her score?(b) What percentage were above?
Answer:
a) 91% of the scores were at or below her score.
b) 9% of the scores were above her score.
Step-by-step explanation:
When a value V is said to be in the xth percentile of a set, x% of the values in the set are lower than V and (100-x)% of the values in the set are higher than V.
In this problem, we have that:
Angela scored on the 91st percentile.
(a) What percentage of the scores were at or below her score?
Since Angela scored on the 91st percentile, 91% of the scores were at or below her score.
(b) What percentage were above?
Since Angela scored on the 91st percentile, 9% of the scores were above her score.
Angela scored higher than or equal to 91% of her peers, meaning 9% scored above her.
Explanation:If Angela scored in the 91st percentile on her general aptitude test, that means she scored higher than or equal to 91% of the other scores. Therefore, (a) 91% of scores were at or below Angela's score. As for (b), if you subtract her percentile from 100%, you can determine that 9% of scores were above hers.
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