Answer:
Rocket will land on the ground in 3 seconds.
Step-by-step explanation:
Given:
[tex]A(t) = -16t^2 + 32t + 48[/tex]
where [tex]t[/tex] ⇒ time in seconds since launch.
[tex]A(t)[/tex] ⇒ altitude of the rocket after reaching the ground.
we need to find the time at which rocket will land on the ground.
Solution:
Now we can say that;
altitude of the rocket after reaching the ground will be equal to 0.
So;
[tex]A(t)=0[/tex]
Now Substituting [tex]A(t)=0[/tex] in given expression we get;
[tex]0=-16t^2+32+48[/tex]
Now we take -16 common we get;
[tex]0=-16(t^2-2t-3)[/tex]
Now Dividing both side by -16 we get;
[tex]\frac0{-16}=\frac{-16(t^2-2t-3)}{-16}\\\\0=t^2-2t-3[/tex]
Now factorizing the above equation we get;
[tex]0=t^2-3t+t-3\\\\0=t(t-3)+1(t-3)\\\\0=(t+1)(t-3)[/tex]
Now we will find 2 values of t by substituting each separately.
[tex]t+1=0 \ \ \ Or \ \ \ \ t-3 =0\\\\t =-1 \ \ \ \ \ \ Or \ \ \ \ \ \ t=3[/tex]
Now we get 2 values of t one positive and one negative.
Now we know that time cannot e negative hence we will discard it and consider positive value of t.
Hence Rocket will land on the ground in 3 seconds.
Angie has some red and blue beads. 40% of her beads were red. When she lost 50 blue beads, the number of blue beads was reduced by 1/3 its original number of beads. How many beads did Angie have in the end?
Answer:
Step-by-step explanation:
Let x represent the total number of red and blue beads
Angie has some red and blue beads. 40% of her beads were red. This means that the total number of red beads is
40/100 × x = 0.4x
The number of blue beads would be
x - 0.4x = 0.6x
When she lost 50 blue beads, the number of blue beads was reduced by 1/3 its original number of beads. This means that
0.6x - 50 = 0.6x - 0.6x/3
0.6x - 50 = 0.6x - 0.2x = 0.4x
0.6x - 0.4x = 50
0.2x = 50
x = 50/0.2 = 250
The number of blue beads that Angie had initially is
0.6 × 250 = 150
The number of blue beads that Angie has left is
150 - 50 = 100 beads
The number of beads that Angie has in the end is
250 - 50 = 200 beads
An aluminum beam was brought from the outside cold into a machine shop where the temp. was held at 65 F. After 10 minutes the beam warmed up to 35 F and after another 10 minutes, its temp was 50 F. Use Newton's Law of cooling to estimate the beam's initial temp.
Answer:
5° F
Step-by-step explanation:
According to Newton's law of cooling, the rate of change is proportional to the difference between the temperature and the ambient temperature.
dT/dt = k (T − T₀)
Solving this by separating the variables:
dT / (T − T₀) = k dt
ln (T − T₀) = kt + C
T − T₀ = Ce^(kt)
T = T₀ + Ce^(kt)
We're given that T₀ = 65.
T = 65 + Ce^(kt)
At t = 10, T = 35.
35 = 65 + Ce^(10k)
-30 = Ce^(10k)
At t = 20, T = 50.
50 = 65 + Ce^(20k)
-15 = Ce^(20k)
Squaring the first equation:
900 = C² e^(20k)
Dividing by the second equation:
-60 = C
Therefore:
T = 65 − 60e^(kt)
At t = 0:
T = 65 − 60e^(0)
T = 5
The initial temperature is 5° F.
What is the 6th term in the sequence described by the following recursive formula?
a1=9
an=an−1−4
Answer:
[tex] a_{6} = - 11[/tex]
Final answer:
The 6th term in the recursive sequence with the first term 9 and each subsequent term decreasing by 4 from the previous term is -11.
Explanation:
To find the 6th term in the sequence described by the recursive formula given:
a1=9
an=an-1−4
We will apply the formula recursively to determine each term up to the 6th term.
First term (a1): 9
Second term (a2): a1 − 4 = 9 − 4 = 5
Third term (a3): a2 − 4 = 5 − 4 = 1
Fourth term (a4): a3 − 4 = 1 − 4 = -3
Fifth term (a5): a4 − 4 = -3 − 4 = -7
Sixth term (a6): a5 − 4 = -7 − 4 = -11
Therefore, the 6th term in the sequence is -11.
The build a dream construction company has plans for two models of the homes they build, model a and model b. The model a home requires 18 single windows and 3 double windows. The model b home requires 20 single windows and 5 double windows. A total of 1,800 single windows and 375 double windows have been ordered for the developments. Write and solve a system of equations to represent this situation. Define your variables. Interpet the solution of the linear system in terms of the problem situation
Answer:
a = 50 houses
b = 45 houses
Step-by-step explanation:
Given
Number of houses called Model A = a
Number of houses called Model B = b
Total of single windows = 1800
Total of double windows = 375
then we have the system of equations
18a + 20b = 1800 (I)
3a + 5b = 375 (II)
Solving the system by whatever method we prefer, we obtain
(I) a = (1800 - 20b)/18
then (II)
3((1800 - 20b)/18) + 5b = 375
⇒ 300 - (10/3)*b + 5b = 375
⇒ (5/3)*b = 75
⇒ b = 45 houses
then
a = (1800 - 20*45)/18
⇒ a = 50 houses
50 model A homes and 45 model B homes will be built.
To solve the problem, let's define our variables:
A = number of model A homesB = number of model B homesWe then create the following system of equations based on the given information:
1. For single windows:
18A + 20B = 1800
2. For double windows:
3A + 5B = 375
We can solve this system using the substitution or elimination method.
Step-by-Step Solution:
Multiply the second equation by 4 to align the coefficients of A:12A + 20B = 1500Subtract the modified second equation from the first equation:(18A + 20B) - (12A + 20B) = 1800 - 15006A = 300A = 50Substitute A = 50 back into the second original equation:3(50) + 5B = 375150 + 5B = 3755B = 225B = 45The solution to the system is A = 50 and B = 45, meaning that the construction company plans to build 50 model A homes and 45 model B homes.
The army bus has 12 seats on one side. Two soldiers can sit in each seat. If five seats are reserved for equipment, how many buses will they need for 1120 soldiers
Answer:
80 buses will be required for 1120 soldiers.
Step-by-step explanation:
Given:
Number of seats on One side = 12 seats.
Now Given:
five seats are reserved for equipment.
So we can say that;
Number of seats used by soldiers = [tex]12-5=7\ seats[/tex]
Number of soldiers on Each seat =2
So we will now find number of soldiers on each bus.
number of soldiers on each bus is equal to Number of seats used by soldiers multiplied by Number of soldiers on Each seat.
framing in equation form we get;
number of soldiers on each bus = [tex]7\times2 = 14\ soldiers[/tex]
Now we know that;
For 14 soldiers = 1 bus
So 1120 soldiers = Number of buses required for 1120 soldiers.
By Using Unitary method we get;
Number of buses required for 1120 soldiers = [tex]\frac{1120}{14} =80[/tex]
Hence 80 buses will be required for 1120 soldiers.
A large container has a maximum capacity of 64 ounces. The container is filled with 8 ounces less than it's maximum capacity. What is the percent of its capacity is the large container filled?
Answer:
87.5%
Step-by-step explanation:
64:64-8
64:56
56/64 *100 = 87.5%
Jessica plots the data points relating the amount of money she needs to repay a loan and the number of months she has been making payments.
A 2-column table with 5 rows. The first column is labeled months with entries 6, 12, 18, 24, 30. The second column is labeled amount to repay (dollar sign) with entries 2,700; 2,110; 1,110; 870; 220.
A graph shows months labeled 5 to 60 on the horizontal axis and amount to repay (dollar sign) on the vertical axis. A line shows a downward trend.
She calculates two regression models. Which is true?
The linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months.
The linear model better represents the situation because according to the exponential model, the repayment amount will never be 0.
The exponential model better represents the situation because the amount she owes decreases by about the same amount every 6 months.
The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
Answer:
The answer is A on E2020!!
A) The linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months.
Answer:
a was right
Step-by-step explanation:
Ashton has a piece of string that is 520 centimeters long. He cuts the string into 8 equal peices and uses 6 of the pieces for a project. How many centimeters of string does Ashton use for his project?
Answer:Ashton used 390 centimeters of string for his project.
Step-by-step explanation:
The total length of the piece of string that Ashton has is 520 centimeters.
He cuts the string into 8 equal pieces. This means that the length of each string would be
520/8 = 65 centimeters.
If he 6 of the pieces for a project, it means that the number of centimeters of string that Ashton used for his project would be
65 × 6 = 390 centimeters
Which equation does not support the fact that polynomials are closed under multiplication?
−1⋅−1=1
1/x⋅x=1
1⋅x=x
1/3⋅3=1
Answer:
The second choice:
[tex]\large\boxed{\large\boxed{1/x\cdot x=1}}[/tex]
Explanation:
The closure property on an operation means that the operation between two elements of a set produce one element of the same set.
In this case, the operation is multiplication and the set is the polynomials.
Then, the closrue property is that the multiplication of two polynomials will always produce a polynomial.
Since, [tex]1/x[/tex] is not a polynomial, the equation [tex]1/x\cdot x=1[/tex] does not support the fact that polynomials are closed under multiplication.
Your math test has 38 questions and is worth 200 points. This test consists of multiple-choice questions worth 4 points each and open-ended questions worth 20 points each. How many of each type of question are there?
Answer: the number of multiple-choice questions in the math test is 35 and the number of open-ended questions in the math test is 3
Step-by-step explanation:
Let x represent the number of multiple-choice questions in the math test.
Let y represent the number of open-ended questions in the math test.
The math test has 38 questions. It means that
x + y = 38
This test consists of multiple-choice questions worth 4 points each and open-ended questions worth 20 points each. The total number of points is 200. It means that
4x + 20y = 200 - - - - - - - - - -1
Substituting x = 38 - y into equation 1, it becomes
4(38 - y) + 20y = 200
152 - 4y + 20y = 200
- 4y + 20y = 200 - 152
16y = 48
48/16
y = 3
Substituting y = 3 into x = 38 - y, it becomes
x = 38 - 3 = 35
Why would there be different published values for the normal range of a particular measurement? why do these values have to be updated periodically?
Answer and Step-by-step explanation:
For general measurements, different people or organizations normally make slightly different measurements. Measurements are never a hundred percent accurate.
The published values are usually updated because in the modern world of discoveries, change is the only constant thing. As new discoveries roll in or not, it becomes necessary to update the current standards; no change in the updated value means the old standards hold, and any change is also updated in the published update.
For health standards/ranges, Different countries have different standards of health
And this requires regular updating because standards of health changes frim time to time.
There was a country concert held at the park. For every 4 men there were 5 women that went to the concert. If 81 people attended the concert, how many men and how many women each attended the concert?
Answer:
36 men, 45 women
Step-by-step explanation:
36 men and 45 women attended the concert
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given,
There was a country concert held at the park.
For every 4 men there were 5 women that went to the concert
81 people attended the concert.
We need to find how many men and how many women each attended the concert.
4x+5x=81
9x=81
Divide both sides by 9
x=9
So 4(9)=36
5(9)=45
Hence, 36 men and 45 women attended the concert
To learn more on Equation:
https://brainly.com/question/10413253
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The following data reflect the number of customers who return merchandise for a refund on Monday. Note these data reflect the population of all 10 Mondays for which data are available. 40 12 17 25 9 46 13 22 16 7Based on these data, what is the standard deviation?
Answer:
The standard deviation of given data is 12.36
Step-by-step explanation:
We are given the following data in the question:
40, 12, 17, 25, 9, 46, 13, 22, 16,7
Formula:
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{207}{10} = 20.7[/tex]
Sum of squares of differences =
372.49 + 75.69 + 13.69 + 18.49 + 136.89 + 640.09 + 59.29 + 1.69 + 22.09 + 187.69 = 1528.1
[tex]\sigma = \sqrt{\dfrac{1528.1}{10}} = 12.36[/tex]
Thus, the standard deviation of given data is 12.36
Final answer:
The standard deviation of the provided data set (number of merchandise returns on Mondays) is approximately 14.8.
Explanation:
To calculate the standard deviation of the provided data set (40, 12, 17, 25, 9, 46, 13, 22, 16, 7), we will follow these steps:
Find the mean of the data set.Subtract the mean from each data point and square the result.Calculate the sum of the squared differences.Divide this sum by the number of data points (since we have the population standard deviation).Take the square root of the result from step 4.Let's apply these steps:
The mean (average) is (40 + 12 + 17 + 25 + 9 + 46 + 13 + 22 + 16 + 7) / 10 = 20.7Squared differences (rounded to two decimal places) would be: (40 - 20.7)², (12 - 20.7)², (17 - 20.7)², and so on.The sum of these squared differences is approximately 2190.1Divide by the number of points, 2190.1 / 10 = 219.01The square root of 219.01 is approximately 14.8Therefore, the standard deviation is about 14.8.
At Jefferson High School, there are 325 students who drive to school 400 students that ride the bus to school. The number of students who drive to school is % of the number of students who ride the bus to school.
Answer:
81.25%
Step-by-step explanation:
Given: There are 325 students who drive to school.
There are 400 students that ride the bus to school.
Now, finding percentage of the number of student who drive to school over number of students who ride the bus to school.
Percentage of student who drive to school= [tex]\frac{325}{400} \times 100[/tex]
⇒ Percentage of student who drive to school= [tex]\frac{325}{4}[/tex]
⇒ Percentage of student who drive to school= [tex]81.25\%[/tex]
Hence, 81.25% is the percent of students who drive to school on the number of students who ride the bus to school.
what is the length of the missing side of the triangle? 24,66 29.15 26.5 30.6
Answer:
24.66
Step-by-step explanation:
You use the Pythagorean Theorem and do 27^2 -11^2= x
Which equation best shows that 55 is a multiple of 11? Choose 1 answer: (Choice A) A 55 = 44 + 1155=44+1155, equals, 44, plus, 11 (Choice B) B 11\times5=5511×5=5511, times, 5, equals, 55 (Choice C) C 11\div55 = 511÷55=511, divided by, 55, equals, 5 (Choice D) D 55 - 11 = 4455−11=44
Answer:
11x5=55
Step-by-step explanation:
The equation that best shows that 55 is a multiple of 11 is 11 × 5 = 55.
Explanation:The equation that best shows that 55 is a multiple of 11 is Choice B: 11 × 5 = 55.
To determine if a number is a multiple of another number, we need to check if the first number can be divided evenly by the second number without any remainder. In this case, 55 can be divided evenly by 11 because 11 × 5 equals 55.
The other choices, A, C, and D, do not represent the relationship of 55 being a multiple of 11.
Sammy and kaden went fishing using live shrimp as bait. Sammy brought 8 more shrimp than kaden brought. When they combined their shrimp they had 32 shrimp altogether. How many shrimp did each boy bring
Answer: kaeden brough 12 sammy brought 20
Step-by-step explanation:
i know that 16+16=32 and
15+17=32 and
14+18=32 and
13+19= 32 and
12+20=32. it’s gonna be 12 and 20 because those numbers add up to 32 and are 8 away from each other. since sammy had more fish he brought 20 and kaeden brough 12.
Final answer:
Kaden brought 12 shrimp and Sammy brought 20 shrimp to their fishing trip. This was determined by solving an algebraic equation set up based on the given information.
Explanation:
The question involves Sammy and Kaden, who went fishing and brought live shrimp as bait. Sammy brought 8 more shrimp than Kaden. Together, they had 32 shrimp. To solve for how many shrimp each boy brought, we can set up algebraic equations. Let the number of shrimp Kaden brought be represented by k, hence Sammy brought k + 8 shrimp. Adding together the shrimp both boys brought gives us:
k + (k + 8) = 32
Simplifying the equation:
2k + 8 = 32
2k = 32 - 8
2k = 24
k = 24 / 2
k = 12
Kaden brought 12 shrimp and Sammy brought 12 + 8 shrimp, which equals 20 shrimp.
So, Kaden brought 12 shrimp, and Sammy brought 20 shrimp.
Plato's Foods has ending net fixed assets of $84,400 and beginning net fixed assets of $79,900. During the year, the firm sold assets with a total book value of $13,600 and also recorded $14,800 in depreciation expense. How much did the company spend to buy new fixed assets?
a. -$23,900
b. $3,300
c. $32,900
d. $36,800
e. $37,400
A worker is handling the four of a rectangle room that is 12 feet by 15 feet.The tiles are square with side lengths 1 1/3 feet. How many tiles are needed to cover the entire floor? Show your reasoning.Show your reasoning
Answer:
Step-by-step explanation:
The measure of the floor of the rectangular room that is 12 feet by 15 feet. The formula for determining the area of a rectangle is expressed as
Area = length × width
Area of the rectangular room would be
12 × 15 = 180 feet²
The tiles are square with side lengths 1 1/3 feet. Converting 1 1/3 feet to improper fraction, it becomes 4/3 feet
Area if each tile is
4/3 × 4/3 = 16/9 ft²
The number of tiles needed to cover the entire floor is
180/(16/9) = 180 × 9/16
= 101.25
102 tiles would be needed because the tiles must be whole numbers.
In the data set below, find the lower quartile, the median, and the upper quartile 6 9 9 4 4 3 2 2 6 8
Answer:
Median = 5
Lower Quartile = 2.5
Upper Quartile = 8.5
Step-by-step explanation:
- First of all, you need to order the numbers from lowest to greatest: 2 2 3 4 4 6 6 8 9 9
- Then, you will find at the number that sits exactly at the middle of this set of numbers. Because this is a set of numbers that has 10 numbers, you will have to look at the two middle points (4 and 6) and divide them by 2 (essentialy finding the average).
4+6=10, 10/2 = 5 = Mean
- To find the upper and lower quartiles, you basically have to find the medians of the set of numbers that are below and above the central median
- So, for the lower quartile, the set of numebrs is: 2 2 3 4. The median sits between 2 and 3, so we have to find the average of those: 2+3=5, 5/2=2.5
- For the upper quartile, the set of numbers is 6 8 9 9. The median is the average of 8+9. So 8+9=17, 17/2 = 8.5
Lower Quartile (Q1) = 3 Median (Q2) = 5 Upper Quartile (Q3) = 8
First, arrange the data in ascending order:
2, 2, 3, 4, 4, 6, 6, 8, 9, 9
The median is the middle value of the sorted data set. Since there are 10 values (an even number), the median will be the average of the 5th and 6th values.
The 5th value is 4
The 6th value is 6
Median (Q2) = (4 + 6) / 2
= 5
The lower quartile is the median of the lower half of the data set (excluding the overall median). For this data set, the lower half is:
2, 2, 3, 4, 4
Since there are 5 values in this half, the lower quartile is the 3rd value.
Lower Quartile (Q1) = 3
The upper quartile is the median of the upper half of the data set (excluding the overall median). For this data set, the upper half is:
6, 6, 8, 9, 9
Since there are 5 values in this half, the upper quartile is the 3rd value.
Upper Quartile (Q3) = 8
A rain gutter is made from sheets of aluminum that are 16 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross- sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area?
Final answer:
The depth of the gutter that will maximize its cross-sectional area is 4 inches, and the maximum cross-sectional area that allows the greatest amount of water to flow is 32 square inches.
Explanation:
To determine the depth of the gutter that will maximize its cross-sectional area, we first need to assume that turning up the edges of the aluminum sheet at right angles will form a rectangular cross-section. If the width of the aluminum is 16 inches and 'x' represents the depth of the gutter (the height of the sides when bent), the width of the base of the gutter will be 16 - 2x (since both sides are turned up).
This means the cross-sectional area 'A' in square inches will be A = x(16 - 2x). This is a quadratic equation and can be expanded as A = -2x^2 + 16x. To find the maximum area, we need to find the vertex of this parabola, which occurs at x = -b/(2a), where 'a' is the coefficient of x^2 and 'b' is the coefficient of 'x'.
In our case, a = -2 and b = 16, so the depth that maximizes the area is x = -16/(2*(-2)) = 4 inches. Therefore, the maximum cross-sectional area is A = 4(16 - 2*4) = 4(8) = 32 square inches.
The depth of the gutter that will maximize its cross-sectional area is 16 inches, and the maximum cross-sectional area is[tex]\( 768 \)[/tex] square inches.
To solve this problem, we will use calculus to find the depth of the gutter that maximizes its cross-sectional area. We will start by defining the dimensions of the gutter and then use the derivative of the area function to find the critical points. Finally, we will determine which of these critical points gives the maximum area.
Let's denote the depth of the gutter as [tex]\( x \)[/tex]inches. Since the width of the aluminum sheets is 16 inches, the base of the gutter will also be 16 inches. When the edges are turned up to form right angles, the gutter will have a rectangular base and two rectangular sides.
The area of the base of the gutter is [tex]\( 16x \)[/tex]. The area of each side is [tex]\( x^2 \),[/tex] and there are two sides, so the total area of the sides is[tex]\( 2x^2 \).[/tex] Therefore, the total cross-sectional area [tex]\( A \)[/tex]of the gutter is the sum of the area of the base and the areas of the two sides:
[tex]\[ A(x) = 16x + 2x^2 \][/tex]
To find the depth that maximizes the area, we need to take the derivative of [tex]\( A(x) \)[/tex] with respect to[tex]\( x \)[/tex]and set it equal to zero:
[tex]\[ A'(x) = \frac{d}{dx}(16x + 2x^2) = 16 + 4x \][/tex]
Setting [tex]\( A'(x) \)[/tex] equal to zero gives us the critical points:
[tex]\[ 16 + 4x = 0 \][/tex]
[tex]\[ 4x = -16 \][/tex]
[tex]\[ x = -4 \][/tex]
Since the depth of the gutter cannot be negative, we discard[tex]\( x = -4 \)[/tex]and realize that we need to consider the physical constraints of the problem. The actual critical point occurs at the endpoint of the domain of [tex]\( x \),[/tex]which is[tex]\( x = 0 \)[/tex](no gutter) or[tex]\( x = 16 \)[/tex] (the gutter's width). Since[tex]\( x = 0 \)[/tex]gives a minimum area (no gutter at all), the maximum area must occur at [tex]( x = 16 \).[/tex]
Now, we calculate the cross-sectional area at [tex]\( x = 16 \)[/tex]
[tex]\[ A(16) = 16(16) + 2(16)^2 \][/tex]
[tex]\[ A(16) = 256 + 2(256) \][/tex]
[tex]\[ A(16) = 256 + 512 \][/tex]
[tex]\[ A(16) = 768 \][/tex]
Therefore, the maximum cross-sectional area of the gutter is[tex]\( 768 \)[/tex]square inches when the depth is equal to the width, which is 16 inches.
What is the surface area of a cube that has a side length of 2.2 meters? Use the formula S A = 6 s squared, where SA is the surface area of the cube and s is the length of each side
Answer:
Step-by-step explanation:
surface area=6*2.2²=6×4.84=29.04 m²
Answer:
29.04
Step-by-step explanation:
just took the test
The quadratic mean of two real numbers x and y equals p (x 2 y 2)/2. By computing the arithmetic and quadratic means of different pairs of positive real numbers, formulate a conjecture about their relative sizes and prove your conjecture.?
Answer:
The quadratic mean of 2 real positive numbers is greater than or equal to the arithmetic mean.
Step-by-step explanation:
x and y Quadratic Mean Arithmetic mean
3 and 3 3 3
2 and 3 2.55 2.5
3 and 6 4.74 4.5
2 and 5 3.8 3.5
2 and 17 12.1 9.5
18 and 28 23.5 23
10 and 48 34.7 29
The quadratic mean is always greater than the arithmetic mean except when x and y are the same.
When the difference between the pairs is small the difference in the means is also small. As that difference increases the difference in the means also increases.
So we conjecture that the quadratic mean is always greater than or equal to the arithmetic mean.
Proof.
Suppose it is true then:
√(x^2 + y^2) / 2) ≥ (x + y)/2 Squaring both sides:
(x ^2 + y^2) / 2 ≥ (x + y)^2 / 4 Multiply through by 4:
2x^2 +2y^2 ≥ (x + y)^2
2x^2 +2y^2 >= x^2 + 2xy + y^2
x^2 + y^2 >= 2xy.
x^2 - 2xy + y^2 ≥ 0
(x - y)^2 ≥ 0
This is true because the square of any real number is positive so the original inequality must also be true.
The quadratic mean of two real numbers, x and y, is given by the formula sqrt((x^2 + y^2)/2). A conjecture can be made that the quadratic mean is greater than or equal to the arithmetic mean for positive real numbers. This conjecture can be proved using the AM-QM inequality and algebraic manipulations.
The quadratic mean of two real numbers, x and y, is given by the formula:
Q(x, y) = sqrt((x^2 + y^2)/2)
To formulate a conjecture about the relative sizes of the arithmetic and quadratic means of different pairs of positive real numbers, we can compare the two means for various pairs of numbers. Based on observations, it can be conjectured that the quadratic mean is always greater than or equal to the arithmetic mean for positive real numbers.
To prove the conjecture, we can use the AM-QM inequality, which states that the quadratic mean is greater than or equal to the arithmetic mean:
Q(x, y) >= A(x, y)
Where Q(x, y) is the quadratic mean and A(x, y) is the arithmetic mean.
Let's consider two positive real numbers, a and b:
Q(a, b) = sqrt((a^2 + b^2)/2)
A(a, b) = (a + b)/2
Now, we need to prove that Q(a, b) >= A(a, b):
Start with the inequality:(a^2 + b^2)/2 >= (a + b)/2
Multiply both sides of the inequality by 2:a^2 + b^2 >= a + bCombine like terms:a^2 - a + b^2 - b >= 0
Factor the expression:(a^2 - a) + (b^2 - b) >= 0
Factor out 'a' and 'b':a(a - 1) + b(b - 1) >= 0
Since 'a' and 'b' are positive numbers, both terms on the left side of the inequality are non-negative.a(a - 1) >= 0
The above inequality is true for all positive 'a' values, and the same holds for 'b'.Therefore, Q(a, b) >= A(a, b), which confirms the conjecture that the quadratic mean is always greater than or equal to the arithmetic mean for positive real numbers.
Learn more about Quadratic Mean here:https://brainly.com/question/35432156
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In a certain game, you pick a card from a standard deck of 52 cards. If the card is a heart, you win. If the card is not a heart, the person replaces the card to the deck, reshuffles, and draws again. The person keeps repeating that process until he picks a heart, and the point is to measure how many draws did it take before the person picks
Answer:32 cards
Step-by-step explanation:
just did it
Brandon eats half the amount of pie that Mollie eats.Yuki eats four times as much pie as Brandon. Mollie eats 1/4 of the pie. How much pie does Anna eat?
Answer: Anna eats 1/8 of pie
Step-by-step explanation: Let total pie =1
Mollie eats 1/4 of pie
Brandon eats half the amount of pie that Mollie eats i.e 1/2 of (1/4)
⇒1/8
Yuki eats 4 times as much as pie as Brandon i.e 4*(1/8)
⇒1/2
Total pie eaten by Mollie +Brandon+Yuki = 1/4+1/8+1/2
⇒7/8
Therefore Anna eats (1-7/8)
⇒ 1/8 of pie
The box plot shows information about the marks scored in a test. Nobody gained 30, 48 or 70 marks. 120 students gained less than 70 marks. How many students gained more than 48 marks?
80 students gained more than 48 marks.
In the given box plot, we have the following information:
- The lowest test score is **10**, and the highest is **100**.
- The 25th percentile (Q1) is **30**, the median (Q2) is **48**, and the 75th percentile (Q3) is **70**.
- No student scored exactly **30**, **48**, or **70** marks.
- **120 students** scored less than **70** marks.
Let's analyze this:
1. The interquartile range (IQR) contains the middle **50%** of the data. Since the median (Q2) is **48**, we know that **50%** of the students scored more than **48** marks.
2. We are interested in how many students scored above **48**. Since **50%** scored more than **48**, the remaining **50%** scored less than or equal to **48**.
3. Given that **120 students** scored less than **70**, we can infer that **75%** of the students scored below **70** (since each region contains **25%** of the data).
4. Therefore, **25%** of the students scored between **48** and **70** (the region between Q2 and Q3).
5. To find out how many students scored more than **48**, we look at the region above Q2. Since there are **2 regions** above Q2, each containing **40 students** (since 120 students = 75%), the total number of students who scored more than **48** is:
[tex]\(2 \times 40 = 80\)[/tex]
Therefore, 80 students gained more than 48 marks.
The number of students who gained more than 48 marks is : 60
Using the information in the boxplot ;
48 marks = median Total number of students = 120The median represents 50% of the data .
The number above the median value can be calculated thus ;
50% × number of studentsNow we have :
50% × 120 = 60
Hence, the number of students who gained more than 48 marks is : 60
what is 155.78=2.95h+73.18
a. 28
b. 36
c. 3.6
d. none of these
Answer:
a. 28
Step-by-step explanation:
Given:
[tex]155.78=2.95h+73.18[/tex]
We need to evaluate given expression to find the value of 'h'.
Solution:
[tex]155.78=2.95h+73.18[/tex]
Now first we will apply Subtraction property of equality and subtract both side by 73.18 we get;
[tex]155.78-73.18=2.95h+73.18-73.18\\\\82.6=2.95h[/tex]
Now we will use Division property of equality and divide both side by 2.95 we get;
[tex]\frac{82.6}{2.95}=\frac{2.95h}{2.95}\\\\h=28[/tex]
Hence After evaluating given expression we get the value of 'h' as 28.
Solve the system using elimination.
2x+8y = 6
3x -8y = 9
You would do this:
2x+8y=6
+ 3x-8y=9
5x=15
x=3
2x+8y=6
2(3)+8y=6
6+8y=6
8y=0
y=0
So x=3 and y=0
Very simple way to do that. Hope it helped.
Answer: x = 3, y = 0
Step-by-step explanation:
The given system of equations is expressed as
2x+8y = 6 - - - - - - - - - - - - - -1
3x -8y = 9- - - - - - - - - - - - - -2
We would eliminate y by adding equation 1 to equation 2. It becomes
2x + 3x = 6 + 9
5x = 15
Dividing the left hand side and the right hand side of the equation by 5, it becomes
5x/5 = 15/5
x = 3
Substituting x = 3 into equation 1, it becomes
2 × 3 + 8y = 6
6 + 8y = 6
Subtracting 6 from the left hand side and the right hand side of the equation, it becomes
6 - 6 + 8y = 6 - 6
8y = 0
Dividing the left hand side and the right hand side of the equation by 8, it becomes
8y/8 = 0/8
y = 0
HELP!! i dont understand this math question and need help
Answer:
52.2 ft
Step-by-step explanation:
Triangle JSV is similar to triangle HTV so you have the proportion ...
JS/SV = HT/TV
JS/(36 ft) = (5.8 ft)/(4 ft) . . . . . . . fill in the given values
JS = (36 ft)(5.8/4) = 52.2 ft . . . . multiply by 36 ft
The height of the wall is 52.2 ft.
We know that m<HVT = m<JVS because the mirror projects equal angles. We can claim this about the angle theta.
tan(θ) = 5.8/4
θ = [tex]tan^{-1}(5.8/4)=55.4[/tex] degrees approx.
So, we want sin theta in the other triangle. Luckily, we also know that...
cos(55.4°) x hypotenuse = 36
hypotenuse = 63.4 ft approx.
So we can find the height by evaluating...
sin(55.4°) x 63.4 = 52.2 ft
answer: 52.2 ft
If you invested $500 at 5% simple interest for 2 years, how much interest do you earn? Show work and answer in complete sentences to earn full credit.
If you invest $500 at 3% compounded monthly for 2 years, how much interest you do earn? Show work and answer in complete sentences to earn full credit.
Which would you rather do?
Answer:
$50
$30.45
Simple interest.
Step-by-step explanation:
If I invested $500 at 5% simple interest for 2 years, then the amount of interest that I will get will be calculated by the simple interest formula as
[tex]I = \frac{Prt}{100} = \frac{500 \times 5 \times 2}{100} = 50[/tex] dollars.
Now, if I invest $500 at 3% compounded monthly for 2 years, then the amount of compound interest will be calculated by the compound interest formula as
[tex]I = P(1 + \frac{r}{100})^{t} - P = 500(1 + \frac{3}{100})^{2} - 500 = 30.45[/tex] dollars.
So, I will prefer to invest in simple interest as the interest there is more. (Answer)