SSD Provided Links O RATIONAL EXPRESSIONS Simplifying a ratio of factored polynomials: Linear factors Simplify. (18(3x-4))/(6(x-8)(3x-4)) You may leave the numerator and denominator of your answer in

Step-by-step explanation:

[tex]4k + 12 = - 36 \\ 4k = - 36 - 12 \\ 4k = - 48 \\ k = - \frac{48}{4} \\ k = - 12[/tex]

#hope it's helpful to you

Answer:

[tex]x^2+5x+4+\dfrac{3x-8}{x^2+5x-4}[/tex]

Step-by-step explanation:

The dividend is the polynomial which has to be divided.

The divisor is the expression by which the dividend is divided.

Given:

Following the steps of long division, divide the given dividend by the divisor:

[tex]\large \begin{array}{r}x^2+5x+4\phantom{)}\\x^2+5x-4{\overline{\smash{\big)}\,x^4+10x^3+25x^2+3x-24\phantom{)}}}\\{-~\phantom{(}\underline{(x^4+\phantom{(}5x^3-\phantom{(}4x^2)\phantom{-b)))))))))}}\\5x^3+29x^2+3x-24\phantom{)}\\-~\phantom{()}\underline{(5x^3+25x^2-20x)\phantom{)))..}}\\4x^2+23x-24\phantom{)}\\-~\phantom{()}\underline{(4x^2+20x-16)\phantom{}}\\3x-8\phantom{)}\end{array}[/tex]

The quotient q(x) is the result of the division.

The remainder r(x) is the part left over:

The solution is the quotient plus the remainder divided by the divisor:

[tex]\implies q(x)+\dfrac{r(x)}{b(x)}=\boxed{x^2+5x+4+\dfrac{3x-8}{x^2+5x-4}}[/tex]

The mean of the list  is 7

The standard deviation of the list is 4/69

The linear regression equation is y = 227.5 + 0.65*x
The percentile rank on the Math portion for a student with a Verbal percentile rank of 80% is approximately 73.5%.

2a) The mean of the list can be computed by adding all of the numbers in the list together and dividing by the number of items in the list.

Mean = (9+2+5+4+12+10)/6 = 42/6 = 7

2b) The standard deviation of the list can be computed by finding the difference between each number in the list and the mean, squaring these differences, finding the average of these squared differences, and then taking the square root of this average.

Standard deviation = sqrt(((9-7)^2 + (2-7)^2 + (5-7)^2 + (4-7)^2 + (12-7)^2 + (10-7)^2)/6) = sqrt(22) = 4.69

3a) The linear regression equation can be found using the formula:

y = b0 + b1*x

Where b0 is the y-intercept and b1 is the slope. The slope can be found using the formula:

b1 = r*(SDy/SDx)

Plugging in the given values:

b1 = 0.80*(85/105) = 0.65

The y-intercept can be found using the formula:

b0 = meany - b1*meanx

Plugging in the given values:

b0 = 570 - 0.65*525 = 227.5

So the linear regression equation is:

y = 227.5 + 0.65*x

To predict the Math score of a student who receives a 720 on the Verbal portion of the test, plug in x = 720 into the equation:

y = 227.5 + 0.65*720 = 693

So the predicted Math score is 693.

3b) To find the percentile rank on the Math portion for a student with a Verbal percentile rank of 80%, use the formula:

z = (x-mean)/SD

Where z is the z-score, x is the score, mean is the mean of the scores, and SD is the standard deviation of the scores.

Plugging in the given values for the Verbal scores:

z = (x-525)/105

Solving for x:

x = 105*z + 525

Since the Verbal percentile rank is 80%, the z-score is 0.84 (from a z-table). Plugging this into the equation:

x = 105*0.84 + 525 = 613.2

So the Verbal score corresponding to the 80th percentile is 613.2.

To find the Math score corresponding to this Verbal score, plug in x = 613.2 into the linear regression equation:

y = 227.5 + 0.65*613.2 = 623.6

So the Math score corresponding to the 80th percentile on the Verbal portion is 623.6.

To find the percentile rank on the Math portion for this score, use the formula:

z = (x-mean)/SD

Plugging in the given values for the Math scores:

z = (623.6-570)/85 = 0.63

Using a z-table, the corresponding percentile rank is approximately 73.5%.

So the percentile rank on the Math portion for a student with a Verbal percentile rank of 80% is approximately 73.5%.

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Both products finish at the same time, which is D) 42 hours later.

The factory produces Product A every 6 hours and Product B every 21 hours.

If they started at the same time, they will finish at the same time after the lowest common multiple of the two intervals, which is 42 hours.

Therefore, the answer is D. 42 hours.

LCM is the short form for “Least Common Multiple.” The least common multiple is defined as the smallest multiple that two or more numbers have in common.

For example: Take two integers, 2 and 3.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20….

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 ….

6, 12, and 18 are common multiples of 2 and 3. The number 6 is the smallest. Therefore, 6 is the least common multiple of 2 and 3.

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The product between f(x) and g(x) is equal to -6x² + 37x - 35.

The product of two functions, (fg)(x), is defined as f(x)*g(x). Therefore, to find the product of the given functions, we simply need to multiply them together:

(fg)(x) = (6x -7)(5 - x)

Using the distributive property, we can simplify this expression:

(fg)(x) = (6x)(5) - (6x)(x) - (7)(5) + 7(x)

(fg)(x) = 30x - 6x² - 35 + 7x

Combining like terms, we get:

(fg)(x) = -6x² + (30x + 7x) - 35

(fg)(x) = -6x² + 37x - 35

So the product of the two functions is (fg)(x) = -6x² + 37x - 35.

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Answer:$880

Step-by-step explanation:

1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3
4    8   12  16  20 24 28 32 36 40 44 48 52 56 60 64

55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55
so to explain 1 cheese pizza to 3 pepperonis(1/3) that is 4 pizzas together, and we do that until 64. If pepperoni pizza is 15 each, we time that by 3 which is 45, plus the 10 for cheese pizza which makes it 55. we do 55 dollars until we equal the 64 pizzas, which is 16 together. so 55 x 16=880 dollars selling pizza

14 : 25 best represents the number of unshaded squares to total squares. The solution has been obtained by using arithmetic operations.

The four mathematical operations that result from dividing, multiplying, adding, and subtracting are quotient, product, sum, and difference.

We are given a figure in which some squares are shaded and rest are unshaded.

Number of total squares = 10 × 10 = 100

Number of shaded squares = 44

So, Number of unshaded squares = 100 - 44 = 56

Ratio of unshaded squares to total squares is as follows

56 : 100

Reducing to the lowest, we get

14 : 25

Hence, 14 : 25 best represents the number of unshaded squares to total squares.

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The perimeter of Raina's pigpen is 1020 feet.

Perimeter is the total distance around the edge of a two-dimensional shape. It is the sum of the lengths of all sides of the shape. Perimeter is typically measured in units such as inches, feet, meters, or centimeters, depending on the system of measurement being used.

To find the perimeter of the rectangular pigpen, we need to add up the lengths of all four sides. The formula for the perimeter of a rectangle is:

perimeter = 2 * length + 2 * width

Plugging in the values we know, we get:

perimeter = 2 * 325 + 2 * 185

perimeter = 650 + 370

perimeter = 1020

Therefore, the perimeter of Raina's pigpen is 1020 feet.

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Since PNDM got dilated about the origin and formed P' N' D' M', the dilation rule is (x, y)    →    (2x, 2y).

In Geometry, the scale factor of a geometric figure can be calculated by dividing the dimension of the image (new figure) by the dimension of the pre-image (original figure):

Scale factor = Dimension of image (new figure)/Dimension of pre-image(original figure)

Substituting the given parameters into the scale factor formula, we have the following;

Scale factor = Dimension of image/Dimension of pre-image

Scale factor = 4/2

Scale factor, k = 2.

Therefore, the dilation rule is given by:

(x, y)      →    (kx, ky)

(x, y)      →    (2x, 2y)

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Answer:

[tex] \frac{ {a}^{x} }{ {a}^{y} } = {a}^{x - y} [/tex]

It will take 62 minutes for the printer to print 93 cards.

We are given the equation p=1.5m, where p represents the number of cards printed and m represents the time in minutes. We can use this equation to find out how many minutes it will take to print 93 cards.

First, we can substitute p=93 into the equation:

93 = 1.5m

Next, we can solve for m by dividing both sides of the equation by 1.5:

m = 93 / 1.5

m = 62

We can verify this answer by plugging in m = 62 into the original equation and checking that we get p = 93:

p = 1.5m

p = 1.5(62)

p = 93

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The complete question is:

The equation p = 1.5m represents the rate at which the printer prints cards where p represents the number of cards and m represents minutes. How many minutes will it take the printers to print 93 cards?

By Pythagorean theorem, the length of line segment x is approximately equal to 2.609 feet.

In this problem we find a geometric system where line segment x is perpendicular to the hypotenuse of a right triangle. This system can be represented well by Pythagorean theorem:

4.6² = (5.8 - y)² + x²

3.5² = x² + y²

First, eliminate variable x and solve for y:

4.6² = (5.8 - y)² + (3.5² - y²)

4.6² = 5.8² - 10.6 · y + y² + 3.5² - y²

4.6² = 5.8² - 10.6 · y + 3.5²

10.6 · y = 5.8² + 3.5² - 4.6²

y = 2.333

Second, determine variable x:

x = √(3.5² - 2.333²)

x = 2.609

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Sure, here are the answers to Exercises 4.4 1 for Propositions 1.5 through 1.8:

Proposition 1.5: If two triangles have two sides and the included angle of one equal to two sides and the included angle of the other, then the triangles are congruent.

Does the proposition hold on a sphere?

No, the proposition does not hold on a sphere. A counterexample is two triangles on a sphere with the same side lengths and angles but with different shapes.

If it does not, give a counterexample and briefly explain what goes wrong.

The problem is that the definition of congruence for plane triangles does not apply to triangles on a sphere since the side lengths and angles are not independent. Two triangles with the same side lengths and angles can have different shapes on a sphere.

If it does hold, does Euclid's proof work? If it does not, briefly explain what goes wrong.

Euclid's proof assumes that the side lengths and angles of the triangles are independent, which is not the case on a sphere. Therefore, his proof does not work for triangles on a sphere.

Proposition 1.6: If two triangles have two angles and a side of one equal to two angles and a side of the other, then the triangles are congruent.

Does the proposition hold on a sphere?

No, the proposition does not hold on a sphere. A counterexample is two triangles on a sphere with the same angle measures and side lengths but with different shapes.

If it does not, give a counterexample and briefly explain what goes wrong.

The problem is that the definition of congruence for plane triangles does not apply to triangles on a sphere since the side lengths and angles are not independent. Two triangles with the same angle measures and side lengths can have different shapes on a sphere.

If it does hold, does Euclid's proof work? If it does not, briefly explain what goes wrong.

Euclid's proof assumes that the angle measures and side lengths of the triangles are independent, which is not the case on a sphere. Therefore, his proof does not work for triangles on a sphere.

Proposition 1.7: If two triangles have two sides and an angle of one equal to two sides and an angle of the other, then the triangles are congruent.

Does the proposition hold on a sphere?

No, the proposition does not hold on a sphere. A counterexample is two triangles on a sphere with the same side lengths and angle measures but with different shapes.

If it does not, give a counterexample and briefly explain what goes wrong.

The problem is that the definition of congruence for plane triangles does not apply to triangles on a sphere since the side lengths and angles are not independent. Two triangles with the same side lengths and angle measures can have different shapes on a sphere.

If it does hold, does Euclid's proof work? If it does not, briefly explain what goes wrong.

Euclid's proof assumes that the side lengths and angles of the triangles are independent, which is not the case on a sphere. Therefore, his proof does not work for triangles on a sphere.

Proposition 1.8: If two triangles have three sides of one equal to three sides of the other, then the triangles are congruent.

Does the proposition hold on a sphere?

No, the proposition does not hold on a sphere. A counterexample is two triangles on a sphere with the same side lengths but with different shapes.

If it does not, give a counterexample and briefly explain what goes wrong.

The problem is that the definition of congruence for plane triangles does not apply to triangles on a sphere

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Answer:34

Step-by-step explanation:

$a = -8$ and $b = \frac{-520}{49}$

The matrix equation you provided is:

$$\left[\begin{array}{rr} 5 & a \\ 4 & -5 \end{array}\right]\left[\begin{array}{rr} b & -4 \\ -1 & 5 \end{array}\right]=\left[\begin{array}{rr} 6 \\ -7 \end{array}\right]$$

We can solve for the missing values by using the matrix equation $A \cdot B = C$. In this case, we have $A \cdot B = C$, where $A$ is the matrix on the left, $B$ is the matrix on the right, and $C$ is the matrix on the far right.

We can solve for $a$ and $b$ by using the inverse of $A$. To calculate the inverse of $A$, we use the formula: $$A^{-1} = \frac{1}{\det(A)}\left[\begin{array}{cc} -5 & a \\ 4 & 5 \end{array}\right]$$

The determinant of $A$ is $5 \cdot -5 - 4 \cdot a = -25 - 4a$, so $$A^{-1} = \frac{1}{-25 - 4a}\left[\begin{array}{cc} -5 & a \\ 4 & 5 \end{array}\right]$$

We can then calculate $b$ by multiplying the inverse of $A$ by the matrix $C$. $$\begin{align*}
A^{-1} \cdot C &= \frac{1}{-25 - 4a}\left[\begin{array}{cc} -5 & a \\ 4 & 5 \end{array}\right]\left[\begin{array}{rr} 6 \\ -7 \end{array}\right]\\
&= \frac{1}{-25 - 4a}\left[\begin{array}{rr} -5\cdot 6 + a\cdot (-7) \\ 4 \cdot 6 + 5 \cdot (-7) \end{array}\right]\\
&= \frac{1}{-25 - 4a}\left[\begin{array}{rr} -30 - 7a \\ 24 - 35 \end{array}\right]
\end{align*}$$

Solving for $b$, we have $$\begin{align*}
b &= \frac{-30 - 7a}{-25 - 4a}\\
&= \frac{-30 - 7a}{-25 - 4a}\cdot \frac{25 + 4a}{25 + 4a}\\
&= \frac{25\cdot(-30 - 7a) + 4a\cdot(-25 - 4a)}{25^2 + 4a^2}\\
&= \frac{-750 - 175a - 100a - 16a^2}{625 + 16a^2}
\end{align*}$$

Now, we can solve for $a$ by substituting $b$ into the equation $A \cdot B = C$. $$\begin{align*}
A\cdot B &= \left[\begin{array}{rr} 5 & a \\ 4 & -5 \end{array}\right]\left[\begin{array}{rr} b & -4 \\ -1 & 5 \end{array}\right]\\
&= \left[\begin{array}{rr} 5b + a(-4) & 5(-4) + a(5) \\ 4b - 5(-1) & -5(-1) - 5(5) \end{array}\right]\\
&= \left[\begin{array}{rr} -20 - 4a & 25 + 5a \\ 4b - 5 & 25 - 25 \end{array}\right]
\end{align*}$$

Comparing this to the equation $A\cdot B = C$, we can solve for $a$: $$\begin{align*}
6 &= -20 - 4a\\
-7 &= 4b - 5\\
&= 4\left(\frac{-750 - 175a - 100a - 16a^2}{625 + 16a^2}\right) - 5\\
&= \frac{-3000 - 700a - 400a - 64a^2}{625 + 16a^2} - 5\\
&= \frac{-3500 - 700a - 400a - 64a^2}{625 + 16a^2}\\
\end{align*}$$

By comparing this to the equation $6 = -20 - 4a$, we can solve for $a$: $$\begin{align*}
-3500 - 700a - 400a - 64a^2 &= -20 - 4a\\
64a^2 + 400a + 700a + 3520 &= 0\\
a^2 + 5a + 56 &= 0\\
(a + 8)(a + 7) &= 0\\
a &= -8 \;\text{or}\; -7
\end{align*}$$

Thus, the missing values are $a = -8$ and $b = \frac{-520}{49}$.

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-17.

According to the remainder theorem, the remainder when dividing a polynomial f(x) by a linear factor x-a is equal to f(a). In this case, we want to find the remainder when dividing f(x) = -4x^3 + 2x^3 -3x+5 by x - 2, so we need to find f(2).

f(2) = -4(2)^3 + 2(2)^3 -3(2) + 5
f(2) = -4(8) + 2(8) -6 + 5
f(2) = -32 + 16 -6 + 5
f(2) = -17

Therefore, the remainder when dividing f(x) = -4x^3 + 2x^3 -3x+5 by x - 2 is -17.

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The solution to the equation (5.825)^(x-3)=120 is x ≈ 4.76.

To solve the equation (5.825)^(x-3)=120 using inverse operations and algebraic methods, we need to use the following steps:

Step 1: Take the natural logarithm of both sides of the equation to isolate the exponent. This gives us:

ln(5.825)^(x-3) = ln(120)

Step 2: Use the property of logarithms that allows us to move the exponent to the front of the logarithm:

(x-3) ln(5.825) = ln(120)

Step 3: Divide both sides of the equation by ln(5.825) to isolate the variable:

x-3 = ln(120)/ln(5.825)

Step 4: Add 3 to both sides of the equation to solve for x:

x = ln(120)/ln(5.825) + 3

Step 5: Use a calculator to find the value of the natural logarithms and simplify:

x ≈ 4.76

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The number of units that must be sold for the company to earn a profit of $50,000 is 30,000 units.

To determine the number of units that must be sold for the company to earn a profit of $50,000, we need to use the following formula:

Profit = Total Revenue - Total Cost

Where Total Revenue = Selling Price × Number of Units Sold

And Total Cost = Fixed Cost + (Variable Cost × Number of Units Sold)

Plugging in the given values into the formula, we get:

$50,000 = ($12 × Number of Units Sold) - ($70,000 + ($8 × Number of Units Sold))

Simplifying the equation, we get:

$50,000 = $12 × Number of Units Sold - $70,000 - $8 × Number of Units Sold

$50,000 = $4 × Number of Units Sold - $70,000

$120,000 = $4 × Number of Units Sold

Number of Units Sold = $120,000 / $4

Number of Units Sold = 30,000

Therefore, the company must sell 30,000 units to earn a profit of $50,000.

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According to the information, the expressions that would match the descriptions would be both expressions have the same result, so it doesn't matter which description they are associated with. both are the largest and the smallest value.

To find the correct descriptions that match the expressions we must look at the graph. In this case q and n represent two numbers on the number line. In this case, to relate them to a description we must find the number to which they refer:

In accordance with the above, we could say that the descriptions would look like this:

In this case, both expressions have the same result, so it doesn't matter which description they are associated with. both are the largest and the smallest value.

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(5z+2)/(z) is the possible rational expression.

To subtract the two rational expressions, we can combine the numerators and keep the common denominator. The subtraction of the two rational expressions is shown below:

(19z+6)/(3z) - (4z)/(3z) = (19z+6-4z)/(3z)

Simplifying the numerator gives:

(15z+6)/(3z)

We can further simplify the expression by factoring out a common factor of 3 from the numerator:

3(5z+2)/(3z)

The 3 in the numerator and denominator cancel out, leaving us with the final simplified expression:

(5z+2)/(z)

Therefore, the answer is (5z+2)/(z).

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The final answer of values of m and n that make the system of linear equations have infinite number of solutions are m = 11/10 and n = 11/5.

To determine the values of m and n so that the following system of linear equations have infinite number of solutions, we need to make the coefficients of both equations proportional.

This means that the ratio of the coefficients of x, y, and the constant term should be the same for both equations.

Let's start by finding the ratio of the coefficients of x and y for the first equation:

(2m-1)/3 = 3/(n-1)

Cross-multiplying gives us:

3(2m-1) = 3(n-1)

Simplifying:

6m-3 = 3n-3

6m = 3n

Dividing both sides by 3:

2m = n

Now, let's find the ratio of the coefficients of x and the constant term for the first equation:

(2m-1)/(-5) = 3/(-2)

Cross-multiplying gives us:

-10m + 5 = -6

Simplifying:

-10m = -11

Dividing both sides by -10:

m = 11/10

Substituting this value of m back into the equation 2m = n, we get:

2(11/10) = n

n = 22/10

n = 11/5

Therefore, the values of m and n that make the system of linear equations have infinite number of solutions are m = 11/10 and n = 11/5.

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Using the unitary method, we found that Zach's plants grew  60cm in 7 weeks.

The unitary method is a strategy for problem-solving that involves first determining the value of a single unit and then multiplying that value to determine the required value. Therefore, the goal of this method is to establish values in relation to a single unit. Always write the things that need to be calculated on the right side and the things that are known on the left side to simplify things. The unitary approach must be applied whenever we need to determine the ratio of one quantity to another.

Given that the height the plant grew in 2 weeks  = 2 feet

We are asked to convert it into centimetres.

This can be done using the unitary method.

Now, the unitary method can be used to find the value of multiple units when the value of the single unit is known.

Here we are given,

1 ft = 0.3 m

But we need feet and centimetre relation.

So we use the metre and centimetres relation.

1m = 100cm

Using the unitary method,

0.3m = 0.3* 100 = 30 cm

So,

1 feet = 0.3m = 30 cm

Then,

2 feet = 30 * 2 = 60 cm

Therefore using the unitary method, we found that Zach's plants grew 60 cm in 7 weeks.

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Answer:

I believe opposing angles are parallel, so it would be 98

Step-by-step explanation:

The total decline in employment over the 5-year period is 384 employees.

In 2000, a business had 4000 employees. Each year for the next 5 years, the number of employees decreased by 2%. To find the number of employees at the end of the 5-year period, we can use the formula:

Number of employees = Initial number of employees * (1 - percentage decrease) ^ number of years

= 4000 * (1 - 0.02) ^ 5

= 4000 * 0.98 ^ 5

= 4000 * 0.9039

= 3615.6

Therefore, the number of employees at the end of the 5-year period is approximately 3616.

To find the total decline in employment over the 5-year period, we can subtract the final number of employees from the initial number of employees:

Total decline in employment = Initial number of employees - Final number of employees

= 4000 - 3616

= 384

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a) The confidence interval of a 95% confidence interval for the proportion of returns at the store in Maniou province is exactly between 0.3718 and 0.4282

b) Do not reject the null hypothesis. The correct answer is option C.

a) To construct a 95% confidence interval we first need to calculate the sample proportion (p) and the standard error (SE) of the proportion.

p = 12/80 = 0.15
SE = sqrt[(p*(1-p))/n] = sqrt[(0.15*0.85)/80] = 0.0347

Using the Z-score for a 95% confidence interval (1.96), we can calculate the lower and upper bounds of the interval:

Lower bound = p - 1.96*SE = 0.15 - 1.96*0.0347 = 0.083
Upper bound = p + 1.96*SE = 0.15 + 1.96*0.0347 = 0.217

Therefore, the 95% confidence interval for the proportion of returns at the store in Maniou province is between 0.083 and 0.217.

b) To determine if the proportion of returns at the Maniou store is significantly different from the returns for the nation as a whole, we can use a hypothesis test. The null hypothesis is that the proportion of returns at the Maniou store is equal to the national proportion (0.06), and the alternative hypothesis is that the proportions are different.

Using the sample proportion (p) and standard error (SE) calculated above, we can calculate the Z-score:

Z = (p - 0.06)/SE = (0.15 - 0.06)/0.0347 = 2.59

Using a Z-table, we can find the p-value for this Z-score. The p-value is 0.0096, which is less than the significance level of 0.05. Therefore, we can reject the null hypothesis and conclude that the proportion of returns at the Maniou store is significantly different from the returns for the nation as a whole.

In conclusion, the correct answer is option C:
a) The confidence interval is exactly between 0.083 and 0.217
b) Reject the null hypothesis

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Answer:

9

Step-by-step explanation:

The median is the number in the middle of a set of numbers.

Our set of numbers:  1, 1, 2, 3, 7, 9, 10, 10, 15, 17, 19​

In this case the median is 9

The function with the largest rate of change is function a.

The rate of change defines how fast the function grows.

So, the most "vertical" or the one that grows the fastest is the function with the largest rate of change.

By looking at the graph, we can see that the fastest growing (the steepest) one is function a, so that is the correct option.

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Answer:

Step-by-step explanation:

Answer:

[tex]f^{-1}[/tex] = [tex]\frac{x-3}{2}[/tex]

The inverse of a function is just the opposite of the function.