Answer:
(1, -2)
(-8, -2)
Step-by-step explanation:
(1 , -6) and (-8 , -6)
1 - (-8) = 9
we know that the length of the side that we know the vertices is 9
from there we make an equation with the sum of the sides equal to the perimeter
we will have 2 times 9 and 2 times x beacause it is a rectangle
x + x + 9 +9 = 26
2x + 18 = 26
2x = 26 - 18
2x = 8
x = 8/2
x = 4
Now that we know the missing side we just have to add or subtract this value to the coordinate in and of the vertices we have and we will obtain the missing vertices
(1, -6 + 4)
(1, -2)
( -8, -6+4 )
(-8, -2)
The coordinates of the other two vertices of the rectangle are (1, -2) and (-8, -2), found by calculating the length of one side using the given vertices and then applying the rectangle's perimeter to find the length of the adjacent sides.
Explanation:The subject of the question is to find the other two vertices of a rectangle given two of its vertices and the perimeter. We know that the opposite sides of a rectangle are equal in length. So, to solve this, we can use the distance formula to find the length of one side with the two given points (1, -6) and (-8, -6). The length of this side is the absolute value of the difference in the x-coordinates, which is 9 units. Since the perimeter is 26, and this length is 9, the sum of the lengths of the other two sides is 26 - 2*9 = 8 units. Therefore, each of these sides is 4 units long. Because the given points have the same y-coordinate, they lie on a horizontal side of the rectangle, so the other two vertices will have the same x-coordinates as the given ones and will be 4 units vertically away. If we add and subtract 4 units from the y-coordinate of the given points, we get the other two vertices: (1, -6 +4) and (-8, -6 +4). So the coordinates of the other two vertices are: (1, -2) and (-8, -2).
Maria is trying to decide which one of two winter coat she should but the bluecoat usually cost $48 but it's on sale for 25% off the black coat is really cost $56 but it's on sale for 40% off how much less is the sale price of a black coat in the sale price of a bluecoat
The sale price of a black coat is $ 2.4 less than sale price of a blue coat
Solution:
Given that,
Bluecoat usually cost $48 but it's on sale for 25% off
Cost price of blue coat = $ 48
Discount = 25 %
Therefore, discount price is given as:
Discount price = 25 % of 48
[tex]Discount\ price = 25 \% \times 48\\\\Discount\ price = \frac{25}{100} \times 48\\\\Discount\ price = 0.25 \times 48\\\\Discount\ price =12[/tex]
Thus sales price is given as:
Sales price = cost price - discount price
Sales price = 48 - 12
Sales price = 36
Thus sales price of blue coat is $ 36
Black coat is really cost $56 but it's on sale for 40%
Cost price of black coat = $ 56
Discount = 40 %
Therefore, discount price is given as:
Discount price = 40 % of 56
[tex]Discount\ price = 40 \% \times 56\\\\Discount\ price = \frac{40}{100} \times 56\\\\Discount\ price = 0.4 \times 56\\\\Discount\ price =22.4[/tex]
Thus sales price is given as:
Sales price = cost price - discount price
Sales price = 56 - 22.4
Sales price = 33.6
Thus sales price of black coat is $ 33.6
How much less is the sale price of a black coat in the sale price of a blue coat
Sales price of blue coat - sales price of black coat = 36 - 33.6 = 2.4
Thus sale price of a black coat is $ 2.4 less than sale price of a blue coat
Answer:
2.40
Step-by-step explanation:
A survey found that 3 out of 5 seventh graders have an email account. It there are 315 seventh graders how many would you expect to have an email account?
For each $n \in \mathbb{N}$, let $A_n = [n] \times [n]$. Define $B = \bigcup_{n \in \mathbb{N}} A_n$. Does $B = \mathbb{N} \times \mathbb{N}$? Either prove that it does, or show why it does not.
Answer:
No, it is not.
Step-by-step explanation:
The set [tex] C = \mathbb{N} \times \mathbb{N}[/tex] contains every ordered pair of Natural numbers, while B only contains those pairs in which both values in each entry are the same. Therefore, C is a bigger set than B, but B is not equal to C because for example C contains [tex][1] \times [2] [/tex] and B doesnt because 1 is not equal to 2.
Zak has a bag of cherries. He gave away 18 cherries to tim and 18 cherries to janet. Now he has 25 cherries. H ow many cherries did zack have at the start?
Answer: he had 68 cherries in the start
Step-by-step explanation:
In the beginning, Zak had 61 cherries and he gave away 18 cherries to Tim and 18 cherries to Janet.
What is the equation?The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.
Cherries are in a bag that Zak possesses. Tim and Janet each received 18 cherries from him. He currently has 25 cherries.
Let's assume the number of cherries Zak had at the start "x".
Zak gave away 18 cherries to Tim and 18 cherries to Janet, so he now has x - 18 - 18 = 25 cherries.
You can solve this equation for x by adding 18 + 18 to both sides, which gives you x = 25 + 18 + 18.
This simplifies to x = 61, so Zak had 61 cherries at the start.
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Decide which food truck you would like to purchase (the blue or green food truck) and determine what the total cost will it be to make it fully functional. A local business has decided to donate 3 times as much money we you have saved in order to purchase the food truck. What is the minimum amount you need to save in order to purchase the food truck?
To find out the minimum amount you need to save to buy the food truck (assuming a local business will donate three times your savings), divide the total cost of the truck by four. This is because the saved amount plus the donated amount (which is three times your savings) should be enough to cover the full cost of the truck.
Explanation:The question is asking for the minimum amount that you need to save in order to buy a food truck, assuming a local business will donate 3 times the amount you have saved. To efficiently and accurately calculate this, it's best to start from the total cost of the truck and work backwards.
Firstly, let’s consider the cost to make the food truck fully functional as X (it's not specified whether it's the blue or green food truck). According to the information, the amount saved will be your contribution, and the local business will donate 3 times your saved amount. This means that the total money available to purchase the food truck will be 4 times the saved amount (the saved amount plus the three times donation).
To find out the minimum amount you need to save, we can use the formula:
minimum savings = total cost / 4
This equation shows that the minimum savings needed will be one-fourth of the total cost of the food truck.
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FIRST ANSWER WILL RECEIVE BRAILIEST!!
A truck that can carry no more than 5900 lb is being used to transport refrigerators and upright pianos. Each refrigerator weighs 200 lb and each piano weighs 525 lb. Write and graph an inequality to show how many refrigerators and how many pianos the truck could carry. Will 13 refrigerators and 8 pianos overload the truck?
Answer:
the truck will over load with 13 refrigerators and 8 pianos
Step-by-step explanation:
13 refrigerators is 2600 lbs
8 pianos is 4200 lbs
both added together is 6800 lbs
the truck can only hold 5900 so 13 refrigerators and 8 pianos is too heavy
Answer:
200r +525p ≤ 5900see below for a graphyes, the truck would be overloadedStep-by-step explanation:
The weight of r refrigerators is 200r. The weight of p pianos is 525p. The total weight must not exceed 5900 lb. So, the total weight of r refrigerators and p pianos must satisfy ...
200r +525p ≤ 5900
In addition, we cannot have negative refrigerators or pianos, so we must also have ...
r ≥ 0
p ≥ 0
A graph is attached.
_____
The point (r, p) = (13, 8) is not in the solution space. 13 refrigerators and 8 pianos would overload the truck.
On Friday, the With-It Clothiers sold some jeans at $25 a pair and some shirts at $18 each. receipts for the day totaled $441. On Saturday the store priced both items at $20, sold exactly the same number of each item, and had receipts of $420. How many pairs of jeans and how many shirts were sold each day?
9 pairs of jeans and 12 shirts were sold on each day.
Explanation:Let x be the number of jeans sold and y be the number of shirts sold. From the given information, we can form two equations:
25x + 18y = 441 (equation 1)
20x + 20y = 420 (equation 2)
Multiplying equation 2 by 5, we get 100x + 100y = 2100. Subtracting this equation from equation 1, we get -75x - 82y = -1659. Solving this equation, we find that x = 9 and y = 12.
Therefore, 9 pairs of jeans and 12 shirts were sold on each day.
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PLZ HURRY IT'S URGENT!!
Which equation can be used to find the two numbers whose ratio is 3 to 4 and that have a sum of 35?
3x + 4x = 35
43x=35
34x=35
4x−3x=35
Answer:
3x+4x=35
Step-by-step explanation:
Answer:3x+4x=35
Step-by-step explanation:took the test and got it right
Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 90 degrees occurs at 4 PM and the average temperature for the day is 70 degrees. Find the temperature, to the nearest degree, at 9 AM
Answer:
65°
Step-by-step explanation:
Since the high is given, it is convenient to use that value with a cosine function to model the temperature. The function will be ...
T = A + Bcos(C(x-D))
where A is the average temperature, B is the difference between the high and the average, C is π/12, reflecting the 24-hour period, and D is the time at which the temperature is a maximum. "x" is hours after midnight.
We have chosen to use a 24-hour clock with x=16 at 4 pm. Then the value of T at 9 am is ...
T = 70 +20cos((π/12(9 - 16)) = 70 +20cos(7π/12) ≈ 64.824
The temperature at 9 am is about 65°.
Final answer:
To find the temperature at 9 AM, a sinusoidal function is constructed with an amplitude of 20 degrees, a vertical shift of 70 degrees, and a phase shift adjusted for a high at 4 PM. Plugging in the time value of 9 AM into the function, it is determined that the temperature at 9 AM is approximately the same as the average or midline temperature, which is 70 degrees.
Explanation:
To find the temperature at 9 AM using a sinusoidal function, we first identify essential characteristics of the function. The maximum temperature (high) of 90 degrees occurs at 4 PM (which we'll take as 16 hours on a 24-hour clock) and the average temperature for the day is 70 degrees, which is also the midline of the sinusoidal function. Since this is a typical daily temperature cycle, we assume that the minimum temperature occurs 12 hours after the maximum temperature. Thus, the temperature will have a periodicity of 24 hours. The amplitude of the temperature variation will be the difference between the high temperature and the average temperature (which is 20 degrees in this case).
The sinusoidal function can be written in the form:
T(t) = A · sin(B(t - C)) + D
Where:
C is the phase shift (16 hours for 4 PM)
Using this information, the sinusoidal function for temperature throughout the day is:
T(t) = 20 · sin((2π/24)(t - 16)) + 70
We then plug in t = 9 (for 9 AM), and calculate the temperature:
T(9) ≈ 20 · sin((2π/24)(9 - 16)) + 70
Which gives us, to the nearest degree:
Temperature at 9 AM ≈ 70 degrees.
Since the value of the sine function ranges from -1 to 1, at 9 AM (7 hours before the maximum temperature at 4 PM), the sine value would be negative indicating that the temperature is rising from the minimum towards the average. However, due to the characteristics of sine, the temperature at 9 AM will actually be the same as the temperature at 19 hours (7 PM), which is also 70 degrees.
Sarah described the following situation:
When fertilizer was added to one plant and nothing was added to another plant, there was a noticeable difference in the color of the leaves of the plants.
Which of the following best describes the situation?
This is an example of correlation because the fertilizer causes the plants to change color.
This is an example of causation because the application of fertilizer caused the plant to improve its leaf color.
This is an example of correlation because one plant is being fertilized and the other is not.
This is an example of causation because the leaves on both plants change color.
Answer:
B. This is an example of causation because the application of fertilizer caused the plant to improve its leaf color.
Answer:
B
Step-by-step explanation:
What is the area of this triangle.
Answer:
13 square units
Step-by-step explanation:
You could calculate the length of each side using Pythagorean theorem, then use Heron's formula to find the area. But there's an easier way: find the area of the rectangle that contains the triangle, and subtract the areas of the smaller triangles in the corners.
The area of the rectangle is 4 × 7 = 28.
The area of the upper left triangle is ½(2)(4) = 4.
The area of the upper right triangle is ½(3)(5) = 7.5.
The area of the lower right triangle is ½(1)(7) = 3.5.
So the area of the triangle is:
28 − 4 − 7.5 − 3.5 = 13
Mary had a link of yarn 4 1/3 inches long she cut it into two pieces for her art project one of the pieces was 2 1/2 inches long what was the length of the second piece inches
Answer: the length of the second piece is 1 5/6 inches
Step-by-step explanation:
Mary had a link of yarn 4 1/3 inches. Converting 4 1/3 inches to improper fraction, it becomes 13/3 inches.
She cut it into two pieces for her art project. One of the pieces was 2 1/2 inches long. Converting 2 1/2 inches to improper fraction, it becomes 5/2 inches.
Therefore, the length of the second piece would be
13/3 - 5/2 = (26 - 15)/6 = 11/6 inches.
Converting to mixed fraction, it becomes
1 5/6 inches
Express each of these mathematical statements using predicates, quantifiers, logical connectives, and mathematical operators.
a. The product of two negative real numbers is positive.
b. The difference of a real number and itself is zero.
Answer:
a. a × b > 0 ∀ a,b ∈ R : a,b < 0
b. a - a = 0 ∀ a ∈ R
Step-by-step explanation:
a. Let a and b be the numbers. Since it says product of two numbers is greater than zero, we write a × b > 0. Since a and b are real numbers, we write a,b ∈ R where ∈ denotes element of a set and R is the set of real numbers. We then use the connective ∀ which denotes "for all" to join a × b > 0 with a,b ∈ R. So, we write a × b > 0 ∀ a,b ∈ R. Since a and b are negative, we write a,b < 0. We now use the connective : which denotes "such that" to combine a × b > 0 ∀ a,b ∈ R with a,b < 0 to give a × b > 0 ∀ a,b ∈ R : a,b < 0. So, the expression is
a × b > 0 ∀ a,b ∈ R : a,b < 0
b. Let a be the number. Since we are looking for a difference, we write a - a. Since it is equal to zero, we write a - a = 0. Since a is an element of real numbers,R, we write a ∈ R, where ∈ denotes "element of". So, a ∈ R denotes a is an element of real numbers R. We combine these two expressions with the connective ∀ which denotes "for all" to give a - a = 0 ∀ a ∈ R. So, the expression is
a - a = 0 ∀ a ∈ R
The mathematical statements given are translated into logical expressions with predicates, quantifiers, logical connectives, and mathematical operators. The first statement is written as ∀x∀y ((x < 0 ∧ y < 0) → P(x, y)), and the second as ∀x (D(x, x)).
The mathematical statements can be expressed using predicates, quantifiers, logical connectives, and mathematical operators as follows:
For the statement 'The product of two negative real numbers is positive':
Let the predicate P(x, y) represent 'the product of x and y is positive', where x and y are real numbers. Then, the statement can be written as:
∀x∀y ((x < 0 ∧ y < 0) → P(x, y))
This translates to: 'For all real numbers x and y, if x and y are both negative, then the product of x and y is positive.'.
For the statement 'The difference of a real number and itself is zero':
Let D(x, y) be a predicate that states 'the difference of x and y is zero'. Then, the statement can be formulated as:
∀x (D(x, x))
Which reads as: 'For all real numbers x, the difference of x and itself is zero.'.
Therefore, The first statement is written as ∀x∀y ((x < 0 ∧ y < 0) → P(x, y)), and the second as ∀x (D(x, x)).
Write an explicit formula for the arithmetic sequence 15.6, 15, 14.4, 13.8,..., and then find the 32nd term.
Answer: the 32nd term is - 3
Step-by-step explanation:
The formula for determining the nth term of an arithmetic sequence is expressed as
Tn = a + d(n - 1)
Where
a represents the first term of the sequence.
d represents the common difference.
n represents the number of terms in the sequence.
From the information given,
a = 15.6
d = 15 - 15.6 = 14.4 - 15 = - 0.6
n = 32
The explicit formula for the arithmetic sequence is
Tn = 15.6 - 0.6(n - 1)
We want to determine the value of the 32nd term, T32. Therefore,
T32= 15.6 - 0.6 (32 - 1)
T32 = 15.6 - 18.6
T32 = - 3
In the image below, DE ∥ BC. Find the measure of EC. Set up a proportion and solve for the measure. Show your work and label your answer. PLEASE HELP ME !!
Answer:
EC = 1 ft
Step-by-step explanation:
DE // BC and AC and AD are transversal lines
∴ ∠E≅∠C ⇒ corresponding angles are congruent
∠D≅∠B ⇒ corresponding angles are congruent
∠A≅∠A ⇒ Reflexive property
∴Δ ADE is similar to ΔABC by AA postulate
So, The corresponding sides are in proportion.
[tex]\frac{AC}{AE} = \frac{AB}{AD}[/tex]
AE = 4 ft , AD = 8 ft , AB = 8 + 2 = 10 ft
AC = AE * AB/AD = 4*10/8 = 40/8 = 5 ft
EC = AC - AE = 5 - 4 = 1 ft
So, the Length of EC = 1 ft.
During an auto accident, the vehicle's air bags deploy and slow down the passengers more gently than if they had hit the windshield or steering wheel. According to safety standards, the bags produce a maximum acceleration of 60g, but lasting for only 36 ms (or less).
solution:
maximum acceleration produce by bag =60 g
time to come to stop = 36 ms
applying equation
[tex]V=u-at[/tex]
inserting values
[tex]0=u-(60)(36*10^-^3)[/tex]
[tex]u=(600*36)/1000=21.6 m/s\\[/tex]
now to find distance of penctration:
[tex]S=ut-1/2at^2[/tex]
inserting values
[tex]S= (21.6)(36*10^-^3)-1/2(600)(0.036)^2[/tex]
[tex]S=0.38m[/tex]
hence distance traveled by person before coming to rest is 0.38 m
The number of people estimated to vote in an election was 7,000. The actual number of people who voted was 5,600
Answer:
A. 25% high
B. 12.5% decrease
Step-by-step explanation:
A. The estimate relative to the actual turnout was ...
7000/5600 = 1.25
The estimate was 25% high.
__
B. Relative to the previous election, the turnout was ...
5600/6400 = 0.875 = 1 - 0.125
The percentage decrease from the previous election was 12.5%.
Das has two bags of sweets each bag contains only lime and strawberry sweets there is 20 sweets in each bag in the first bag for every one lime there are 3 strawberry in the second bag there are 2 lime for every 3 strawberry how many more lime were in the second bag than in the first
Answer:
3
Step-by-step explanation:
A binomial probability experiment is conducted with the given parameters. Use technology to find the probability of x successes in the n independent trials of the experiment n=8, p=0.6, x<4
The probability of having fewer than 4 successes (x < 4) in 8 independent trials with a success probability of 0.6 is 0.1758.
We have,
The binomial probability formula:
[tex]P(x) = (^nC_x) p^x (1-p)^{n-x}[/tex]
Where:
P(x) is the probability of x successes
n is the number of trials
p is the probability of success
nCx is the binomial coefficient, which represents the number of ways to choose x successes from n trials
For the given parameters:
n = 8
p = 0.6
x < 4
For x = 0:
[tex]P(0) = (^8C_0) (0.6^0) (1-0.6)^{8-0}\\= (1) (1) (0.4)^8[/tex]
= 0.0016
For x = 1:
[tex]P(1) = (^8C_1) (0.6^1) (1-0.6)^{8-1}\\= (8) (0.6) (0.4)^7[/tex]
≈ 0.0092
For x = 2:
[tex]P(2) = (^8C_2) (0.6^2) (1-0.6)^{8-2}\\= (28) (0.6^2) (0.4)^6[/tex]
≈ 0.0412
For x = 3:
[tex]P(3) = (^8C_3) (0.6^3) (1-0.6)^{8-3}\\= (56) (0.6^3) (0.4)^5\\= 0.1238[/tex]
Now, sum up these probabilities to find the probability of x < 4:
P(x < 4) = P(0) + P(1) + P(2) + P(3)
≈ 0.0016 + 0.0092 + 0.0412 + 0.1238
≈ 0.1758
Therefore,
The probability of having fewer than 4 successes (x < 4) in 8 independent trials with a success probability of 0.6 is 0.1758.
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Veterinarians often use nonsteroidal anti-inflammatory drugs (NSAIDs) to treat lameness in horses. A group of veterinary researchers wanted to find out how widespread the practice was in the United States. They obtained a list of all veterinarians treating large animals, including horses. They sent questionnaires to all the veterinarians on the list. Such a survey is called a census. The response rate was 40%. Which statement is NOT correct?A.Such a low response rate has the potential for response bias.B. The intended sample consisted of the target population.C. The chance to be selected into the sample was the same for all veterinarians.D.The sample was a volunteer sample.
Answer:
C. The chance to be selected into the sample was the same for all veterinarians
The statement that is NOT correct is D. The sample was a volunteer sample.
Explanation:The statement that is NOT correct is D. The sample was a volunteer sample.
The given scenario describes a census survey, where questionnaires were sent to all veterinarians treating large animals. In a census survey, every member of the target population is included, so there is no sampling involved. Hence, there is no opportunity for the sample to be a volunteer sample. Therefore, option D is the incorrect statement.
A low response rate, as mentioned in option A, can lead to response bias because the respondents who choose to participate may have different characteristics than those who do not respond. The intended sample, as mentioned in option B, was the target population of veterinarians treating large animals. And option C is correct since all veterinarians on the list had an equal chance of being selected as part of the survey.
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How do you do this question?
Answer:
B) ∫₂⁵ ∜x dx
Step-by-step explanation:
The factor is 3/n, so b − a = 3
The expression under the radical is 2 + 3k/n, so a = 2. Therefore, b = 5.
The function is f(x) = ∜x.
Plugging into a definite integral:
∫₂⁵ ∜x dx
Ms. Petrie buys some peaches for $4.95 and some breakfast cereal for $7.85. Ms. Petrie had $50 before she went shopping. How much mone does Ms. Petrie have left after she buys the peaches and cereal?
Answer: she has $37.2 left.
Step-by-step explanation:
Total cost of the peaches that Ms. Petrie bought is $4.95.
She some breakfast cereal for $7.85. The total amount that she spent in buying the peaches and cereals would be
4.95 + 7.85 = $12.8
Ms. Petrie had $50 before she went shopping. Therefore, the amount of money that Ms. Petrie have left after she buys the peaches and cereal would be
50 - 12.8 = $37.2
Sheila began running 6 years ago. She has spent 1,204 on 14 pairs of running shoes during this time. How much, on average, did each pair of shoes cost?
Answer: I’m pretty sure $86
Step-by-step explanation:
If I’m correct it would just be $1,204 divided by 14 pairs of shoes.
a student either knows the answer or guesses. Let 3434 be the probability that he knows the answer and 1414 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1414 . What is the probability that the student knows the answer given that he answered it correctly?
Answer: [tex]\dfrac{12}{13}[/tex]
Step-by-step explanation:
Let A = he known the answer then A' = he guess the answer.
B = he answered it correctly
As per given , we have
[tex]P(A)=\dfrac{3}{4}\ \ ,\ \ P(A')=\dfrac{1}{4}[/tex]
[tex]P(B|A)=1[/tex]
[tex]P(B|A')=\dfrac{1}{4}[/tex]
By Bayes theorem , we have
[tex]P(A|B)=\dfrac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A')P(A')}\\\\ P(A|B)=\dfrac{1\times\dfrac{3}{4}}{1\times\dfrac{3}{4}+\dfrac{1}{4}\times\dfrac{1}{4}}\\\\= \dfrac{12}{13}[/tex]
The probability that the student knows the answer given that he answered it correctly is [tex]\dfrac{12}{13}[/tex] .
Claire traveled 701 miles. She drove 80 miles every day. On the last day of her trip she only drove 61 miles. Write and solve an equation to find the number of days Claire traveled. Explain each step of your problem solving strategy.
Answer:
Claire traveled for 9 days.
Step-by-step explanation:
Given:
Total Distance traveled = 701 miles
Distance traveled each day = 80 miles
Distance traveled on last day = 61 miles
We need to find the number of days Claire traveled.
Solution:
Let the number of days Claire traveled be denoted by 'd'.
Now we can say that;
Total Distance traveled is equal to sum of Distance traveled each day multiplied by number of days and Distance traveled on last day.
framing in equation form we get;
[tex]80d+61=701[/tex]
Now Subtracting both side by 61 using Subtraction Property of Equality we get;
[tex]80d+61-61=701-61\\\\80d = 640[/tex]
Now Dividing both side by 80 we get;
[tex]\frac{80d}{80}=\frac{640}{80}\\\\d=8[/tex]
Hence Claire traveled 80 miles in 8 days and 61 miles on last day making of total 9 days of travel.
Two sets are equal if they contain the
same elements. I.e., sets A and B are equal if
∀x[x ∈ A ↔ x ∈ B].
Notation: A = B.
Recall: Sets are unordered and we do not distinguish
between repeated elements. So:
{1, 1, 1} = {1}, and {a, b, c} = {b, a, c}.
Answer:
Definition: Two sets are equal if they contain the
same elements. I.e., sets A and B are equal if
∀x[x ∈ A ↔ x ∈ B].
Notation: A = B.
Recall: Sets are unordered and we do not distinguish
between repeated elements. So:
{1, 1, 1} = {1}, and {a, b, c} = {b, a, c}.
Definition: A set A is a subset of set B, denoted
A ⊆ B, if every element x of A is also an element of B.
That is, A ⊆ B if ∀x(x ∈ A → x ∈ B).
Example: Z ⊆ R.
{1, 2} ⊆ {1, 2, 3, 4}
Notation: If set A is not a subset of B, we write A 6⊆ B.
Example: {1, 2} 6⊆ {1, 3}
A spring requires 12 J12 J to stretch the spring from 8 cm8 cm to 10 cm10 cm, and an additional 48 J48 J to stretch the spring from 10 cm10 cm to 14 cm14 cm. What is the natural (unstressed) length of the spring?
Answer:
6cm
Step-by-step explanation:
Given
12J to stretch from 8cm to 10cm ---- Expression 1
Additional 48J from 10cm to 14cm ---- Expression 2
Let l represents the length of the spring
Let k represent string constant,
We can then write ( from expression 1)
12 = integral of kx dx
The lower bound being 8 - l
And the upper bound being 10 - l
Integrating kx dx
We have
½kx²
= ½k[x²]
= ½k[(10 - l)² - (8 - l)²].
So, 12 = ½k[(10 - l)² - (8 - l)²] ------ Equation 1
From expression 2, we can write
48 = integral of kx dx
The lower bound being 10 - l
And the upper bound being 14 - l
Integrating kx dx
We have
½kx²
= ½k[x²]
= ½k[(14 - l)² - (10 - l)²].
So, 48 = ½k[(14 - l)² - (10 - l)²] ------ Equation 2
Divide Equation 2 by Equation 1, so we get
48/12 = ½k[(14 - l)² - (10 - l)²] / ½k[(10 - l)² - (8 - l)²] ----- ½k cancel out ½k
So, w have
4 = [(14 - l)² - (10 - l)²] / [(10 - l)² - (8 - l)²]
Recall that a² - b² = (a + b)(a - b).
So,
4 = [(14 - l - 10 + l) (14 - l + 10 - l)] / [(10 - l - 8 + l) (10 - l + 8 - l)]
4 = [(4)(24 - 2l)]/[(2)(18-2l)]
4 = (96 - 8l)/36 - 4l) ---- Multiply both sides by 36 - 4l
4(36 - 4l) = 96 - 8l
144 - 16l = 96 - 8l ----- Collect Like Terms
144 - 96 = 16 - 8l
48 = 8l ----- Divide both sides by l
48/8 = l
6 = l --- Rearrange
l = 6
So, the natural length of the string is 6cm
A linear transformation of the form z = Γx was applied to the data, where Γ is a 2 × 2 matrix. The decision boundary associated with the BDR is now the hyperplane of normal w = (1/ √ 2, −1/ √ 2)T which passes through the origin.
Answer:
Part a: The transformation matrix is the clockwise rotation matrix of π/4.
Part b: The hyperplane would move towards the mean of class 1.
Part c: The distance will remain in the Euclidean Space due to the rotation transformation only.
Step-by-step explanation:
As the complete question is not available, the question is searched online and a reference question is obtained which has 3 parts as follows:
Part a:
The decision boundary after transformation coincides with the line x1 = x2, the two class means must lie on a line that is normal to the decision boundary, i.e. on x1 = −x2. This implies that the transformation matrix Γ is a clockwise rotation transformation of π/4, given as
[tex]\Gamma=\left[\begin{array}{cc}\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\\\frac{-\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{array}\right][/tex]
Part b:
If the prior probability of class 0 was increased after transformation, then the decision boundary of BDR would still have the same normal as before, i.e.,[tex]w=(1/\sqrt{2},-/\sqrt{2})^T[/tex], but move toward the mean of class 1.
Part c:
Noting that
[tex]||\bold{T}_x-\bold{T}_y||^2=(x-y)^T \bold{T}^T\bold{T}(x-y)\\||\bold{T}_x-\bold{T}_y||^2=(x-y)^T(x-y)\\||\bold{T}_x-\bold{T}_y||^2=||x-y||^2[/tex]
This indicates that the distance is still the same and is in Euclidean space. This is due to the fact that rotation transformations does not affect the distances between the points.
Ami decided to score an average of 90 marks in the four subjects - Maths, Physics, Chemistry and Biology. The maximum marks in each paper was 100. She scored 75 in Maths and 95 in Physics. Which of these, if she scores, will ensure that she gets the desired average score?
To get an average score of 90 in four subjects, Ami needs to score a combined total of 190 marks in Chemistry and Biology after having scored 75 in Maths and 95 in Physics.
Explanation:Ami is aiming to achieve an average score of 90 across four subjects, with each subject having a maximum score of 100 marks. To determine the scores that she needs to achieve in Chemistry and Biology (the remaining two subjects), we begin by calculating the total marks she requires for the desired average.
Since the desired average is 90, for four subjects, the total marks needed would be: 90 marks/subject × 4 subjects = 360 marks.
Ami scored 75 in Maths and 95 in Physics, which sums up to: 75 + 95 = 170 marks.
To find out the remaining marks she needs, we subtract the marks she has already scored from the total marks needed for the average:
360 marks (total needed) - 170 marks (already scored) = 190 marks (needed for Chemistry and Biology).
Therefore, to ensure that she gets the desired average score of 90, she would need to score a combined total of 190 marks in Chemistry and Biology. This could be achieved by, for example, scoring 95 in both subjects or any other combination that adds up to 190.
Converges or Diverges: Please help
The equation after the sum symbol is written in terms of ar^k-1
A = 12 and r = 0.7
To see if it converges use the formula a /1-r, if the answer is greater than 1 it converges, if it’s less than 1 it diverges
12 / 1 - 0.7 = 12/0.3 = 40
The answer C. Converges, 40