Water is poured into a clear triangular prism as shown to the left. It takes 5 cubic inches of water to fill the prism to the height of 1 inch. 15. What is the area of the BASE of this prism?​

The product of the expression (a + b) and (a - 2b) will be a² - ab - 2b².

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

The expressions are given below.

(a + b) and (a - 2b)

Then the product of the expression is given as,

⇒ (a + b)(a - 2b)

⇒ a² - 2ab + ab - 2b²

⇒ a² - ab - 2b²

The product of the expression (a + b) and (a - 2b) will be a² - ab - 2b².

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The maximum value of the function f(x) = (x − x²)/(1 + 8x²) is 1/4, and it occurs when x = 1/2.

A function is a mathematical relationship between input and output values.

In this problem, we have a specific function defined as

=> f(x) = (x − x²)/(1 + 8x²).

Let's start by analyzing the expression of the function.

The function is composed of two parts:

=> (x − x²) and (1 + 8x²).

The first part is a simple polynomial expression with degree 2. It represents a parabolic shape with a vertex located at (1/2, 1/4).

The second part, (1 + 8x²), is always positive, so it does not affect the shape of the function.

The maximum value of the function will occur when the first part, (x − x²), is at its maximum. Since this part is a parabolic shape with a vertex at (1/2, 1/4), the maximum value is 1/4.

So, the maximum value of the function

=>  f(x) = (x − x2)/(1 + 8x²) is 1/4.

To find the value of x at which this occurs, we will have to find the x-coordinate of the vertex of the parabolic shape.

This can be done using the formula for the x-coordinate of the vertex of a parabolic equation, which is -b/2a.

Plugging in the values of a and b from the equation (x − x²) gives us

x = 1/2.

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Let P(t)=36(1−e^(−kt))+56 represent the expected core for a student who studied t hour for a test, suppose k=0. 24 so student scored 56% without studying.

If the student don't study then,

It corresponds to t= 0 .

We can evaluate the function with the t=0 so we can have

P(t) = 36(1−e^(−kt)) + 56

were k = 0.24

P(0) = 36 (1 - [tex]e^{0.24(t)}[/tex]) + 56

= 36 (1 - [tex]e^{0.24(0)}[/tex]) + 56

= 36 (1 - 1 ) + 56

=56%

Therefore, Let P(t)=36(1−e^(−kt))+56 represent the expected core for a student who studied t hour for a test is 56%.

The derivative of a function of a real variable in mathematics quantifies the sensitivity of the function's value (output value) to changes in its argument (input value). Calculus's core tool is the derivative. The velocity of an item, for instance, is the derivative of its position with respect to time; it quantifies how quickly the object's position varies as time passes.

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Answer:

-3

Step-by-step explanation:

-x^2-6x+8   using quadratic formula has roots  -3 ± sqrt(17).....the axis of symmetry will be in the middle of these roots at  - 3

Answer:

B is the answer, no explanation needed, this question is easy

Step-by-step explanation:

Albert have left 7/4 cups of cheese left.

Fractions in Math, represent a numerical value that expresses a part of a whole. The whole can be any number, a specific value, or a thing.

Given,

Amount of cheddar = 2/3 cup

Amount of mozzarella = 5/6 cup

Amount of parmesan = 3/4 cup

Total cups of cheese

= cheddar + mozzarella + parmesan

= 2/3 + 5/6 + 3/4

= 8 + 10 + 9 /12

= 27/12

= 9/4

He puts 1/2 cup of cheese in pizza

Cheese left = 9/4 - 1/2 = 9 - 2 / 4 = 7/4

There is 7/4 cups of cheese left.

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The percent of 125 is 25 when it is 20% and 70 is 14 when it is 20%

To determine the percentage, we have to divide the value by the total value and then multiply the resultant by 100.

To calculate the percentage of a number, we need to use a different formula such as:

P% of Number = X, where X is the required percentage.

If we remove the % sign, then we need to express the above formula as;

P/100 * Number = X

What percent of 125 is 25 and percent of 70 is 14?

Given:

To solve this, we need to set up a proportion. We know 25 is a certain percent of 125, and we want to find what that percent is.

So, we set up the proportion 25/125 = x/100, where x is the percent, we are looking for.

Now we can cross-multiply to solve for x.

We get 25 x 100 = 125 x, so x = [tex]\frac{25 \times 100}{125}[/tex], x = 20

Therefore, 25 is 20% of 125.

To solve this, we need to set up a proportion.

We know 14 is a certain percent of 70, and we want to find what that percent is.

So, we set up the proportion 14/70 = x/100, where x is the percent, we are looking for.

Now we can cross-multiply to solve for x.

We get 14 x 100 = 70 x

x = [tex]\frac{1400}{70}[/tex]

So, x = 20.

Therefore, 14 is 20% of 70.

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C. 13 + X = 8

X= 8-13

therefore 8-13 will give

X=-5

Answer:

C would be the answer

To choose the best option for the company, it is important to consider the total amount of interest that will be paid over the life of the loan.

With the 30-year loan, the total interest paid over the life of the loan will be higher than with the 15-year loan, because the interest rate is higher and the loan is paid over a longer period of time.

However, the shorter loan term means that the monthly payments are higher, which could be a financial burden for the company. Therefore, it is important to compare the total interest paid over the life of the loan for both options to determine which is the best option for the company.

Complete question:

A company i planning to purchae a plot of land for $200,000 to build a factory there are two option offered by a lending intitution. The first option is to make a down payment of $200,000 and take a $600,000 30-year mortgage loan with an interest rate of 3.5%. The second option is to make a $100,000 down payment and take a $700,000 15-year mortgage loan with an interest rate of 3.0%.

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Answer:

To calculate the monthly payment for a loan, you can use the following formula: P = L[c(1 + c)^n]/[(1 + c)^n – 1]

Step-by-step explanation:

See the attachment for detail

In geometry, an angle is often measured in units of radians, which is a way to express the size of an angle in terms of the length of the arc it cuts off on the circumference of a circle.

This relationship between an angle and an arc length is given by the formula:

Arc Length = Radius * Angle (in radians)

Rearranging this formula to solve for the angle, we have:

Angle (in radians) = Arc Length / Radius

Plugging in the values for the radius (2.2) and the arc length (12.2), we get:

Angle (in radians) = 12.2 / 2.2 = 5.54

Rounding to the nearest 10th, the angle in radians is approximately 5.5.

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Answer:The rate of speed is the rate at which the total distance is traveled in the time taken.

The Space Explorer B travel 6.67 times faster than Space Explorer A.

What is rate of speed?

The rate of speed is the rate at which the total distance is traveled in the time taken. Rate of speed can be given as,

Here,  is the distance traveled by the object and  is time taken but the object to cover that distance.

Step-by-step explanation:The equation given in the problem is,

Here,  represented the number of miles that the satellite, Space Explorer B, flies in hours.

The satellite, Space Explorer A, flies 24000 miles in 20 hours.

The speed of the Space Explorer A, which flies 24000 miles in 20 hours is,

Thus the Space Explorer A, travels 1200 miles per hour.

As the Space Explorer B, travel is 8000 miles per hour and the Space Explorer A, travels 1200 miles per hour. Thus the ratio of the speed of both is,

Thus the Space Explorer B travel 6.67 times faster than Space Explorer A.

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The first pair of numbers is a decimal number and a large whole number. The second pair is a large whole number and decimal number. The third pair is a smaller whole number and the same decimal as the second pair.

The first pair of numbers is 4.24 and 3150. This means that 4.24 is a decimal number and 3150 is a large whole number. In order to understand this pair of numbers, it is important to know how to read decimal numbers. A decimal number is written with a decimal point that separates the whole number part from the fractional part. The number 4.24 is read as "four and twenty-four hundredths", which means that 4.24 is 4 whole numbers and 24 hundredths. The second pair of numbers is 345 and 4.240. This is a large whole number and another decimal number. In this pair, 345 is the large whole number and 4.240 is the decimal number. To read 4.240, first read the whole number part (4) and then the fractional part (240). This means that 4.240 is read as "four and two hundred and forty thousandths". The third pair of numbers is 45 and 4.240. This is a smaller whole number and the same decimal as the second pair. To read 4.240 in this pair, it is the same as reading it in the second pair. This means that 4.240 is read as "four and two hundred and forty thousandths".

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The zeros are x = -5 and x = 1, and the graph can be seen at the end.

Here we have the quadratic function:

f(x) = (x + 5)*(x - 1)

The function is equal to zero when:

(x + 5)*(x - 1) = 0

And that happens if one of the two factors is zero, so:

x + 5 = 0  ----> x = -5

x - 1 = 0 -----> x= 1

So the function is equal to zero for x = -5 and x = 1, to graph them just graph the points (-5, 0) and (1, 0) on the coordinate axis.

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The circumference of the circle is 18.84 cm. Therefore, option C is the correct answer.

The circumference is the length of any great circle, the intersection of the sphere with any plane passing through its centre. A meridian is any great circle passing through a point designated a pole.

Given that, a circle has a diameter of 6 cm.

We know that, the formula to find the circumference of the circle is 2πr.

Here, radius = 6/2 =3 cm

Now, circumference = 2×3.14×3

= 18.84 cm

Therefore, option C is the correct answer.

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a) On solving the linear equations y = - 6x - 11  and y = 4x + 19, the solutions are x = -3 and  y = 7.

b) The system of equations has one solution.

c) The solution is obtained as x = -3 and y = 7.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

The two linear equations given are -

y = - 6x - 11 .... (1)

y = 4x + 19 .... (2)

Substitute equation (1) in (2) -

- 6x - 11 = 4x + 19

Collect all the like terms -

- 6x - 4x = 19 + 11

-10x = 30

x = 30 / -10

x = -3

Substitute the value of x in equation (1) -

y = - 6(-3) - 11

y = 18 - 11

y = 7

The two equations form a linear graph and intersect at point (-3,7).

So, there is only one solution to the system of equations.

Therefore, the solutions are x = -3 and y = 7.

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Answer:

They are exactly the same size and shape

Step-by-step explanation:

The congruence postulates covered in this lesson are

- Side-Side-Side (SSS)

- Side-Angle-Side (SAS)

- Angle-Side-Angle (ASA)

- Angle-Angle-Side (AAS)

If  company packages puzzles in boxes that are 6 Inches by 4 Inches by 2 inches and the shipping e is a cube with 12-inch sides. The greatest number of puzzle boxes that can be packed into a shipping crate is: 36 puzzle boxes.

Volume of the puzzle box = 6 * 4 * 2

Volume of the puzzle box = 48 cubic inches

Volume of the shipping crate = 12 * 12 * 12

Volume of the shipping crate = 1728 cubic inches

Greatest number of puzzle boxes that can be packed into a shipping crate is:

Greatest number of puzzle boxes = 1728 / 48

Greatest number of puzzle boxes = 36 puzzle boxes

Therefore the Greatest number of puzzle boxes is 36 puzzle boxes.

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Once you have determined what x is, you may calculate how many minutes would be spent on each machine.

A limited set of equations for which we looked for common solutions is known as an equation system. It is also known as an equation set or a collection of simultaneous equations. A group of equations can be categorized in the same way as a single equation.

You must learn more about how many calories each of your two machines burns per minute!

However, you can locate a few facts about your workout goals in this word problem to get things started. You can calculate the ratio of 300 calories in 40 minutes to get the number of calories you should aim for per hour or minute:

For an hour's worth of calories, 300 calories divided by 40 minutes is x calories (you are aiming for 450 calories burned per hour)

7.5 calories per minute, or 300 calories divided by 40 minutes, is another way to look at it.

One online variant of this issue completes the setup in this manner. [WARNING: It's possible that your issue from school is different. In that situation, just use this illustration:

Eight calories per minute for the first machine

6 calories per minute for the second machine

This enables us to construct an equational system. Calorie burn will be the first equation.

8x + 6y = 300

On the machine that burns 8 calories per minute, x represents the number of minutes, and y represents the number of minutes.

Simply put, a second equation demonstrates that the sum of the minutes is 40:

x + y = 40

You can now solve the system using substitution, elimination, or linear combinations. I'll modify the second equation using substitution:

x = 40 - y

Now substituting into the first equation:

8(40 - y) + 6y = 300

320 - 8y + 6y = 300

2y = 20

y = 10

Once you have determined what x is, you may calculate how many minutes would be spent on each machine.

The complete question is:

You have 40 minutes to exercise at the gym, and you want to burn 300 calories total using both machines. How much time should you spend on each machine?

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The standard deviation of the data set will increase by a factor of 10. This is because standard deviation measures how much variation or dispersion exists from the mean of a data set. Adding 10 to every value of the data set will increase the overall spread of the data and thus increase the standard deviation.  

To prove this mathematically, let's consider a data set A with mean M and standard deviation SD. Let's add 10 to every value of the data set to get a new data set B. The mean of set B is M+10 and the standard deviation of set B is SD'.   The formula for SD' is:  

SD':[tex]\sqrt{(x_{1} -(M+10))^{2} +(x_{2} -(M+10))^2+ (x_{3}-(M+10))^2 +...+ (x_{n}-(M+10))^2)/n }[/tex] We can simplify this by substituting the values for the xi:

 SD' =[tex]\sqrt{(((x_{1}-M-10)^2 + (x_{2}-M-10)^2 + (x_{3}-M-10)^2 +...+ (x_{n}-M-10)^2)/n) }[/tex]  Now let's consider the original data set A and its standard deviation SD:

 SD =[tex]\sqrt{(((x_{1}-M)^2 + (x_{2}-M)^2 + (x_{3}-M)^2 +...+ (x_{n}-M)^2)/n) }[/tex] ,We can substitute the values for xi in this formula as well:  

SD= [tex]\sqrt{ (((x_{1}-M-10)^2 + (x_{2}-M-10)^2 + (x_{3}-M-10)^2 +...+ (x_{n}-M-10)^2)/n)}[/tex]  If we compare SD and SD', we can see that they are identical, except that the value of M has been replaced with M+10. This means that when we add 10 to every value of the data set, the standard deviation increases by a factor of 10.

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The nature and scope of the conclusion that a study can reach refers to the limitations and generalizability of the results obtained.

It depends on the research design, methodology, and sample size used. The conclusion can only be applied to the specific population and conditions studied.

The study may also have limitations due to biases or confounding variables that can affect the validity of the conclusion. To have a stronger conclusion, it is important to have a well-designed study with a large and representative sample, and to control for extraneous variables.

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∠1 and ∠5 = Corresponding angles.

∠2 and ∠8 = Same-side exterior angles.

∠4 and ∠5 = Alternate interior angles.

Lines in a plane that are consistently spaced apart are known as parallel lines. Parallel lines don't cross each other.

Given:

Two parallel lines cut by a traversal line.

Then,

∠1 and ∠5 = Corresponding angles.

∠2 and ∠8 = Same-side exterior angles.

∠4 and ∠5 = Alternate interior angles.

Therefore, all the required values are given above.

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Answer:c

Step-by-step explanation:

an equation that represents the given scenario is total charges = 0.13x + 29.99.

Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.

Given, A phone company charges 13 cents per minute and a flat fee of $29.99 each month.

Let "x" be the number of minutes

Since Fix cost to pay = 29.99$

Charge Per minute = 13 cents

Charge for x minutes = 0.13x

Thus,

total charges = 0.13x + 29.99

A. equation:

total charges = 0.13x + 29.99

B.  if you use the phone for 250 minutes a month

So, total charges = 0.13 * 250 + 299

Total charges if you use the phone for 250 minutes = 331.5$

C. y-intercept(where x is zero)

Y-intercept represents that you didn't use the phone the entire month

D. the slope represents

As we can see in the given Equation:

The slope represents the Charge for phone use per minute

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A proportional equation for c in terms of s that matches the total cost in dollars and cents for any number of downloaded songs, s, is C = 10.19s.

A proportional equation is an algebraic equation where the constant of proportionality shows the constant ratio between two variables.

The constant ratio between the variables is also known as the slope or the unit rate.

The formula for a proportional equation is y = kx, where y and x are the variables in the equation and k represents the constant of proportionality.

The unit cost per downloaded song = $10.19

The number of downloaded songs = s

The total cost of the downloaded songs = C

C = 10.19s

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The cost per downloaded song = $10.19

For the given triangles ΔABC ≈ ΔADE, the measurement  of angle ACB is, 87°

Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent. Triangle resemblance is shown here by the symbol "≈"

Given that,

One triangle, ΔABC

one segment DE which is intersecting sides AB & AC

ΔABC has one similar triangle inside, which is ΔADE

Triangle ΔABC and ΔADE similar,

So, we can say ΔABC ≈ ΔADE

And it is known that

for similar triangles, ratio of corresponding angles of both the triangles are equal for similar triangles

so, ∠DAE/∠AED  = ∠CAB/∠ABC

⇒  62°/(11x -2)°  =  62°/(6x + 13)°

⇒  (11x -2)°  = (6x + 13)°

⇒   11x - 6x = 13 + 2

⇒    5x  = 15

⇒      x = 3

So,

∠ADE = 11x - 2 = 11 x 3 - 2 = 31°   =  ∠ABC

It is known that,

Sum of all the interior angles of triangle is 180°

Now,

∠ACB + ∠CAB + ∠ABC = 180°

∠ACB + 62° + 31°  = 180°

∠ACB  = 180° - 93°

           =  87°

Hence, the angle ACB is 87°

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Answer: 19/20

Step-by-step explanation:

Probability of tomato spaghetti = tomato sauce and probability of spaghetti

Probability (tomato sauce) = 1/4
Probability (spaghetti) = 1/5
Probability (tomato spaghetti) = 1/4 x 1/5 = 1/20
Probability of not tomato spaghetti = 1- probability of tomato spaghetti
= 1 - 1/20  = 19/20

The solution to the system is:

(x, y) = (2.236, 1), (-2.236, 1), (2.236, -1), or (-2.236, -1).

This represents two equations hyperbola and ellipse which intersect at exactly 4 points.

The system of equations, as follows:

3x² + y² = 16

x² - y² = 4

We can solve this system of equations by substitution.

Starting with the first equation, we can isolate y²:

y² = 16 - 3x²

Substituting that into the second equation, we get:

x² - (16 - 3x²) = 4

Expanding the right side:

x² - 16 + 3x² = 4

4x² = 4 + 16

4x² = 20

Dividing both sides by 4:

x² = 5

Taking the square root of both sides:

x = ±√5 = ± 2.236

With x found, we can substitute it back into the expression for y²:

y² = 16 - 3x²

y² = 16 - 3(√5)²

y² = 16 - 3(5)

y² = 16 - 15

y² = 1

Taking the square root of both sides:

y = ±1

So the solution to the system is:

(x, y) = (2.236, 1), (-2.236, 1), (2.236, -1), or (-2.236, -1)

This means the two equations are hyperbola and ellipse which intersect at exactly 4 points.

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There is sufficient evidence to support the claim of a greater variance for machine 1

Given: n1=25 , n2​=22

The mean is the sum of all values divided by the number of values:

x1=2.95+3.45+...+3.35+3.12/25≈3.3284

x2​=3.22+3.30+...+3.16+3.33​/22≈3.2782

The variance is the sum of squared deviations from the mean divided by n−1. The standard deviation is the square root of the variance:

s1=√(2.95−3.3284)²+....+(3.12−3.3284)²/25−1≈0.2211

s1​=√25−1(2.95−3.3284)2+....+(3.12−3.3284)2​

​≈0.2211

s2=(3.22−3.3284)2+....+(3.33−3.3284)222−1≈0.0768

s2​=(3.22−3.3284)2+....+(3.33−3.3284)2²/22−1​≈0.0768

Determine the hypotheses:

H0=σ12≤σ22

H1​=σ12​>σ22​

Compute the value of the test statistic:

F=[tex]\frac{S^1_2}{S^2_2}[/tex] =0.2211²/0.0768²≈8.288

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table 4 containing the F-value with dfn=25−1=24 and dfd=22−1=21:

P<0.01

If the P-value is less than the significance level, reject the null hypothesis.

P<0.01⇒ Reject H0

Complete Question:

The variance in a production process is an important measure of the quality of the process. A large variance often signals an opportunity for improvement in the process by finding ways to reduce the process variance. Conduct a statistical test to determine whether there is a significant difference between the variances in the bag weights for the two machines. Use a .05 level of significance. What is your conclusion? Which machine, if either, provides the greater opportunity for quality improvements?

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There is sufficient evidence to support the claim of a greater variance for machine 1

Given: n1=25 , n2​=22

The mean is the sum of all values divided by the number of values:

x1=2.95+3.45+...+3.35+3.12/25≈3.3284

x2​=3.22+3.30+...+3.16+3.33​/22≈3.2782

The variance is the sum of squared deviations from the mean divided by n−1. The standard deviation is the square root of the variance:

s1=√(2.95−3.3284)²+....+(3.12−3.3284)²/25−1≈0.2211

s1​=√25−1(2.95−3.3284)2+....+(3.12−3.3284)2​

​≈0.2211

s2=(3.22−3.3284)2+....+(3.33−3.3284)222−1≈0.0768

s2​=(3.22−3.3284)2+....+(3.33−3.3284)2²/22−1​≈0.0768

Determine the hypotheses:

H0=σ12≤σ22

H1​=σ12​>σ22​

Compute the value of the test statistic:

F= [tex]\frac{S_2^1}{S_2^2}[/tex] =0.2211²/0.0768²≈8.288

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table 4 containing the F-value with dfn=25−1=24 and dfd=22−1=21:

P<0.01

If the P-value is less than the significance level, reject the null hypothesis.

P<0.01⇒ Reject H0

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Complete Question:

The variance in a production process is an important measure of the quality of the process. A large variance often signals an opportunity for improvement in the process by finding ways to reduce the process variance. Conduct a statistical test to determine whether there is a significant difference between the variances in the bag weights for the two machines. Use a .05 level of significance. What is your conclusion? Which machine, if either, provides the greater opportunity for quality improvements?

The cost of a bus in 1984 is =$ 7360 and in 1992 is $28140.and CPI in 1950 is 20.5.

The consumer price index (CPI) is the index that shows how much prices have changed over time. This index mainly focuses on the retailer prices for market players. Usually, this index increases due to inflation.

Part(1)

% change in CPI (1984-70)

[tex]=\frac{36.8-100}{100} \\=-63.2 \ percentage \\=-0.632[/tex]

Cost of bus = 20000(1-0.632)=$ 7360

Part II

% change in CPI (1984-92)

[tex]=\frac{140.7-199}{100}[/tex]

=40.7%

= 0.407

cost of a bus in 1992 = 20,000(1+0.407)

= $28140

part (3)

% change in CPI (1950-84)=% change in price

[tex]\frac{100-x}{x}*100=[[20000-4100]/4100]*100\\100-x=3.878x\\=x=20.5\\the\ cpi\ in\ 1950\ was \ 20.5.[/tex]

The complete question is -

In 1984, a school bus cost $20,000. To answer each of the parts below, assume that the consumer price index (CPI) was 36.8 in 1970, 100 in 1984, and 140.7 in 1992.

Part I: If the price of a school bus increased at the same rate as the CPI from 1970 to 1984, how much did a school bus cost in 1970? By how much did the price of a school bus increase from 1970 to 1984?

Part II: If the price of a school bus increased at the same rate as the CPI from 1984 to 1992, how much did a school bus cost in 1992? By how much did the price of a school bus increase from 1984 to 1992?

Part III: If a school bus cost $4100 in 1950, and if the price of a school bus increased at the same rate as the CPI from 1950 to 1984, what was the CPI to the nearest tenth in 1950?

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