(Chapter 7 Polynomials Questions)
What would be the answer be?

The value of  f(-1) = 1 - (-1)²/₃ = 2/3 and f(1) = 1 - 1²/₃ = 2/3.

To find values c in (-1,1) such that f'(c) = 0, we take the derivative of f(x): f'(x) = -2x/3. Setting f'(c) = 0, we get -2c/3 = 0, which implies that c = 0. Therefore, the only value of c in (-1,1) such that f'(c) = 0 is c = 0.

Rolle's Theorem states that if a function is continuous on a closed interval [a,b], differentiable on the open interval (a,b), and f(a) = f(b), then there exists at least one point c in (a,b) such that f'(c) = 0.

In this case, f(x) satisfies the conditions of Rolle's Theorem on the interval [-1,1]. We have shown that there exists exactly one point c in (-1,1) such that f'(c) = 0, namely c = 0. Therefore, Rolle's Theorem holds true for f(x) on the interval [-1,1].

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Answer:

y <= x+2

Step-by-step explanation:

Finding the curve equation,

the slope is 1 and the y-intercept is 2. Hence,

y = x + 2

Since the thing is under the graph,

y < x + 2

Since it is a solid line,

y <= x + 2

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The limit lim(x→1) (x^a - 1)(x^b - 1) is equal to ab.

To evaluate the limit lim(x→1) (x^a - 1)(x^b - 1), we'll follow these steps:

1. Recognize the given expression: (x^a - 1)(x^b - 1)
2. Apply the limit: lim(x→1) (x^a - 1)(x^b - 1)
3. Factor using the difference of squares: (x - 1)(x^(a-1) + x^(a-2) + ... + 1)(x - 1)(x^(b-1) + x^(b-2) + ... + 1)
4. Cancel out the common factor of (x - 1) in both terms: lim(x→1) (x^(a-1) + x^(a-2) + ... + 1)(x^(b-1) + x^(b-2) + ... + 1)
5. Substitute x = 1 in the remaining expression: (1^(a-1) + 1^(a-2) + ... + 1)(1^(b-1) + 1^(b-2) + ... + 1)
6. Simplify: (1 + 1 + ... + 1)(1 + 1 + ... + 1)
7. Count the number of terms in each parenthesis and multiply them.

Since there are "a" terms in the first parentheses and "b" terms in the second parentheses, the final answer is ab.
So, the limit lim(x→1) (x^a - 1)(x^b - 1) is equal to ab.

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For the value of x = 4/5 (= 0.8) the matrix a = [tex]\left[\begin{array}{cc}5&x\\-6&-5\end{array}\right][/tex] is equal to its own inverse.

The matrix a is given as,

a = [tex]\left[\begin{array}{cc}5&x\\-6&-5\end{array}\right][/tex]

The value of x is such that a is equal to its inverse, that is,

a = [tex]a^{-1}[/tex] ___(1)

Inverse of an matrix, say a, can be calculated using the formula ,

[tex]a^{-1}\\[/tex] = (adjoint of matrix a) / (determinant of matrix a)

Therefore, Adjoint of matrix a = [tex]\left[\begin{array}{cc}-5&-x\\6&5\end{array}\right][/tex]

where,

As element of the adjoint matrix in row 1 and column 1 is cofactor of the matrix a in row 1 and column 1,  

As element of the adjoint matrix in row 1 and column 2 is cofactor of the matrix a in row 1 and column 2,  

As element of the adjoint matrix in row 2 and column 1 is cofactor of the matrix a in row 2 and column 1,

And  as element of the adjoint matrix in row 2 and column 2 is cofactor of the matrix a in row 2 and column 2.

Therefore, determinant of matrix a = (5)(-5) - (-6)(x) = -25 +30x

Thus from the formula of inverse of a matrix we get,

[tex]a^{-1}\\[/tex] = {1/( -25 +30x)} [tex]\left[\begin{array}{cc}-5&-x\\6&5\end{array}\right][/tex]

=[tex]\left[\begin{array}{cc}-5/ (-25 +30x)&-x/ (-25 +30x)\\6/ (-25 +30x)&5/ (-25 +30x)\end{array}\right][/tex] ___(2)

Therefore, equating equation (1) and (2) we get,

[tex]\left[\begin{array}{cc}-5/ (-25 +30x)&-x/ (-25 +30x)\\6/ (-25 +30x)&5/ (-25 +30x)\end{array}\right][/tex] = [tex]\left[\begin{array}{cc}5&x\\-6&-5\end{array}\right][/tex]

⇒ -5/ (-25 +30x) = 5,

-x/ (-25 +30x) = x,

6/ (-25 +30x) = -6,

and 5/ (-25 +30x) = -5

From any one of the above four equation we can equate for the value of x we get,

5/ (-25 +30x) = -5

⇒1/ (-25 +30x) = -1

⇒ 25 - 30x = 1

⇒ 30x = 24

⇒ x =24/30 = 4/5 (=0.8)

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The volume of the solid lying under the plane 8-2x-y is 32 cubic units.

To find the volume of the solid lying under the plane 8-2x-y, you need to use a triple integral.

To find the volume, you need to perform a triple integral over the region, integrating the function 8-2x-y with respect to x, y, and z. First, find the limits of integration for x, y, and z by determining the intersections of the plane with the coordinate axes. The intersections are (4,0,0), (0,8,0), and (0,0,8). Next, set up the triple integral as follows:

∭(8-2x-y)dzdydx, with x ranging from 0 to 4, y ranging from 0 to 8-2x, and z ranging from 0 to 8-2x-y.

Evaluate the integral with respect to z first, then y, and finally x. After evaluating, you will find that the volume of the solid is 32 cubic units.

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One possible absolute value inequality with the solution set 5 is:

| x - 5 | ≤ 0

An absolute value inequality is a type of inequality that involves the absolute value of a variable. The absolute value of a number is its distance from zero, and it is always a non-negative value.

The general form of an absolute value inequality is:

| f(x) | < a

where f(x) is an algebraic expression involving x, and a is a positive number.

An absolute value inequality with the solution set of 5 can be written in the form:

| x - b | ≤ c

where b is the value around which x can vary and c is the maximum distance from b to the boundary of the solution set.

To obtain a solution set of 5, we need to choose b as the midpoint between the two endpoints of the solution set, which is (5 + 5)/2 = 5.

The distance from b to either endpoint of the solution set is 5 - 5 = 0. Therefore, we can choose c to be any value greater than or equal to 0.

One possible absolute value inequality with the solution set 5 is:

| x - 5 | ≤ 0

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The best graph to display the data is C. a bar graph titled favorite topping with the x axis labeled topping and the y axis labeled number of customers, with the first bar labeled chocolate chips going to a value of 17, the second bar labeled hot fudge going to a value of 12, the third bar labeled nuts going to a value of 27, and the fourth bar labeled sprinkles going to a value of 44.

Categorical data is best represented through a bar graph wherein every distinctive category is illustrated by a rectangular bar in correspondence to the frequency or item count. This display method uses the height or length of each rectangle as its basis.

The customer's preferences on their choice of ice cream toppings were subjected to a categorical survey; hence, it deemed suitable for visual illustration via a bar chart. There are four recognizable component ingredients identified in this survey namely: Sprinkles, Nuts, Hot Fudge, and Chocolate Chips. Each component underlies particular statistical value and described discretely hence a bar graph proves fitting to portray this information.

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Answer:

The highlighted part is the answer

Step-by-step explanation:

The volume of the given solid is 12 cubic units.

To evaluate the given double integral, we can first identify it as the volume of a solid. In this case, the integrand is (16-8y), which represents the height of the solid at any given point (x,y) in the region R=[0,1]x[0,1].

Thus, to find the volume of this solid, we need to integrate this height function over the entire region R.

∫ ∫ R (16 − 8y)dA = ∫₀¹ ∫₀¹ (16-8y) dx dy

Evaluating this double integral using iterated integration, we get:

∫₀¹ ∫₀¹ (16-8y) dx dy = ∫₀¹ [16x - 8yx] from x=0 to x=1 dy

= ∫₀¹ (16-8y) dy

= [16y - 4y²] from y=0 to y=1

= (16-4) - (0-0)

= 12

Therefore, the volume of the given solid is 12 cubic units.

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To sketch the vector field F(r) = -r / ||r||^3 in the xy-plane, we can first observe that this is a radial vector field that points towards the origin. As ||r||^3 is the cube of the distance from the origin, the denominator increases much faster than the numerator, causing the lengths of the vectors to decrease as we move away from the origin. Therefore, the first statement "The lengths of the vectors decrease as you move away from the origin. All the vectors point towards the origin" is true.

As for the second statement, "The length of each vector is 1. All the vectors point in the same direction. All the vectors point away from the origin", it is not true for this vector field. The length of each vector depends on the distance from the origin and is not constant. Also, the vectors point towards the origin and not away from it. Therefore, this statement is false.

In summary, the correct answer is: The lengths of the vectors decrease as you move away from the origin. All the vectors point towards the origin.

To sketch the vector field F(r) = -r / ||r||^3 in the xy-plane and determine which statements apply, follow these steps:

1. Recognize that F(r) is a radial vector field with its direction determined by the term -r, which points towards the origin, and its magnitude determined by 1/||r||^3.

2. Notice that as you move away from the origin (increasing the value of ||r||), the magnitude of the vector field decreases because the denominator ||r||^3 increases, making the overall value of the vector field smaller.

3. Observe that all vectors point towards the origin because of the negative sign in the term -r.

4. Since the magnitude of the vector field is determined by 1/||r||^3 and not a constant value, the length of each vector is not 1.

5. As the vector field is radial and determined by the term -r, the vectors do not point in the same direction and do not point away from the origin.

From this analysis, we can conclude that the following statements apply:

- The lengths of the vectors decrease as you move away from the origin.
- All the vectors point towards the origin.

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In a case whereby Mai poured 2.4 L into a partilly filled water now there is 10.4 the best figure that represent this is the second fiqure.

Based on the given information it can be seen that the total volume of the figure is 10.4 which implies that it will take the total volume of of water of 10.4

Considering the second fiqure , it can be deduced that the total volume is 10.4, where one part of the fiqure is X and other bis 2.4, which impies that 10.4 = X + 2.4 which is the expression for the fiqure.

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Prisha has 56 apples and bananas. She has three times as many apples than bananas. Prisha has 42 apples.

To determine how many apples Prisha has, we will use the given information and set up an equation involving the terms apples and bananas.

Let A represent the number of apples and B represent the number of bananas.

According to the problem, A + B = 56.

It's also given that Prisha has three times as many apples as bananas, so A = 3B.

Now we can substitute the expression for A from Step 3 into the equation from Step 2:

3B + B = 56.

Combine the terms with B:

4B = 56.

Divide by 4 to find the value of B:

B = 14.

Now, using the value of B, find the value of A:

A = 3B = 3 × 14 = 42.

So, Prisha has 42 apples.

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To calculate the probability that none of the 60 randomly selected subjects left the treatment center against medical advice, we will use some steps.

Those steps are:
1. Calculate the probability of a single subject not leaving against medical advice.
2. Calculate the probability of all 60 subjects not leaving against medical advice.

Step 1:
There are a total of 55,673 patients, out of which 55,423 did not leave against medical advice. So, the probability of a single subject not leaving against medical advice is:
P(not leaving) = (number of patients not leaving) / (total number of patients)
P(not leaving) = 55,423 / 55,673 ≈ 0.9955

Step 2:
Since the subjects are randomly selected without replacement, we need to adjust the probability for each subsequent selection. However, as the sample size (60) is much smaller than the total number of patients (55,673), the difference in probabilities will be negligible. Therefore, we can assume that the probability for each subject remains approximately the same.

To calculate the probability that none of the 60 subjects left the treatment center against medical advice, we will multiply the probability of each subject not leaving against medical advice:

P(all 60 not leaving) = (P(not leaving))^60
P(all 60 not leaving) = (0.9955)^60 ≈ 0.7409

So, the probability that none of the 60 randomly selected subjects left the treatment center against medical advice is approximately 0.7409 or 74.09%.

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The net monthly salary after saving 1% is R 1072.5

From the question, we have the following parameters that can be used in our computation:

Savings = 1%

Annual salary = R13,000

Using the above as a guide, we have the following:

Monthly salary = Annual salary /Number of months

Substitute the known values in the above equation, so, we have the following representation

Monthly salary = 13000 / 12

Next, we have

Monthly salary = 13000 / 12 * (1 - 1%)

Evaluate

Monthly salary = 1072.5

Hence, the net monthly salary is R 1072.5

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S cannot be a basis for [tex]R^{n }[/tex]

What is Matrix ?

A matrix is a rectangular array of numbers or symbols arranged in rows and columns. Matrices are commonly used in mathematics, physics, engineering, computer science, and other fields to represent systems of linear equations, transformations, and other mathematical objects and operations.

The statement "For each b in  [tex]R^{m }[/tex], the equation Ax - b has a solution" is equivalent to the following statements:

A. The columns of A span [tex]R^{m }[/tex]

B. Each b in [tex]R^{m }[/tex] is a linear combination of the columns of A.

E. The matrix A has a pivot position in each column.

To explain why S cannot be a basis for  [tex]R^{n }[/tex] , we can use the fact that a set of vectors S = {v1, ..., vk} is a basis for  [tex]R^{n }[/tex] if and only if the matrix whose columns are the vectors in S is invertible. In this case, since k < n, the matrix whose columns are the vectors in S cannot be invertible because it has more columns than rows.

Therefore, S cannot be a basis for [tex]R^{n }[/tex].

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Answer:

  a) y = |1/2x -2| +4

  b) y = |1/2x -2| +2

  c) y = |1/2x -3/2| +3

  d) y = |1/2x -5/2| +3

Step-by-step explanation:

You want the equations for the translation of y = |1/2x -2| +3 ...

The transformation of a function required to translate it (right, up) by (h, k) units is ...

  f(x) = f(x -h) +k

For (h, k) = (0, 1), the new function is ...

  y = |1/2x -2| +3 +1

  y = |1/2x -2| +4

For (h, k) = (0, -1), the new function is ...

  y = |1/2x -2| +3 -1

  y = |1/2x -2| +2

For (h, k) = (-1, 0), the new function is ...

  y = |1/2(x -(-1)) -2| +3

  y = |1/2(x+1) -2| +3

  y = |1/2x -3/2| +3

For (h, k) = (1, 0), the new function is ...

  y = |1/2(x -1) -2| +3

  y = |1/2x -5/2| +3

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The statement that is true about the quadrilateral is

It is a rectangle because the opposite sides in a quadrilateral are congruent.

Option C is the correct answer.

We have,

From the given information,

AB = CD = √73

BC = AD = √5

Slope:

BA = CD = 1/2

BC = AD = 8/3

Now,

This means,

AB = CD = congruent

BC = AD = congruent

Now,

The opposite sides in a quadrilateral are congruent.

This means,

The quadrilateral is a rectangle.

Thus,

The statement that is true about the quadrilateral is

It is a rectangle because the opposite sides in a quadrilateral are congruent.

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The first part of the question asks us to complete the table and find the slope of the graph at the given point for two different functions. For the function (3x)^4, the derivative is 12(3x)^3. For the function r(θ) = 4 sin θ at the point (0,0), the slope of the graph is 4.

Let's complete the table and find the slope of the graph at the given point using the terms "derivative" and "slope."

1. Original Function: (3x)^4

Rewrite: (3x)^4
Differentiate: Using the power rule, d/dx[(3x)^4] = 4 * (3x)^(4-1) * d/dx(3x)
Simplify: 4 * (3x)^3 * 3 = 12(3x)^3

2. Function: r(θ) = 4 sin θ, Point: (0, 0)

To find the slope of the graph at the given point, we'll differentiate the function r(θ) with respect to θ.

Differentiate: dr/dθ = d/dθ [4 sin θ] = 4 * d/dθ [sin θ] = 4 * cos θ

Now, let's find the slope at the point (0, 0) by plugging θ = 0 into the derivative:

Slope: 4 * cos(0) = 4 * 1 = 4

So, the slope of the graph at the point (0, 0) is 4. To confirm your results, you can use the derivative feature of a graphing utility.

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The population density for each animal is given as follows:

The population density is calculated as the division of the total population by the total area.

The area for this problem is given as follows:

2.22 million acres = 2,220,000 acres.

Hence the densities are given as follows:

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The number of pages with photos rounded to the nearest whole number is 204 pages.

First, we need to find out how many pages of the album are empty. Since 59% of the album is empty, that means 41% of the album is filled with photos.

To find out how many photos are in the album, we multiply the number of pages by the number of photos per page:

500 pages x 5 photos per page = 2500 photos

To find out how many pages are filled with photos, we need to take 41% of the total number of pages:

500 pages x 0.41 = 205 pages

However, since we're looking for the number of pages with photos rounded to the nearest whole number, we round down to 204 pages. Therefore, each of the 18 students would receive 204/18 = 11.33 pages of photos (rounded to the nearest hundredth).

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Complete Question:

Roy has an album that holds. 500 Each page of the album holds 5 photos. If 59​% of the album is​ empty, how many pages are filled with photos​?

The correct answer for the function  is both injective and surjective and it's proves that the function is bijective.

Given:

[tex]f(x) = \dfrac{5x+1}{x-2}[/tex]

If the function is both injective and surjective, the function is bijective:

Check Injective:

For every value in input in the function, their always exist a different output.

for [tex]x =1[/tex]

[tex]f(x)= \dfrac{5(1)+1}{1-2} \\\\= -6[/tex]

for [tex]x=3[/tex]

 [tex]f(x)= \dfrac{5(3)+1}{3-2} \\\\= 16[/tex]

As value for different output is different, function is Injective;

To check Surjectivity:

Show that for every y ∈ R −{5}, there exists an x ∈ R −{2} such that f (x) = y.

Let y ∈ R − {5}.  find an x ∈ R − {2} such that f (x) = y.

Solve f (x) = y for x.

[tex]\dfrac{5x + 1}{x-2} = y[/tex]

[tex]5x+1=xy- 2y[/tex]

[tex]xy-5x-2y+1=0[/tex]

[tex]x(y-5)-2y+1=0[/tex]

[tex]x=\dfrac{2y-1}{ y-5}[/tex]

[tex]f(x) = y[/tex]

The function is bijective.

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Answer:

C is correct in case of division

Step-by-step explanation:

A is correct in case of addition

B is correct in case of multiplication

Answer:C

Step-by-step explanation: take the exponent value in the numerator  and subtract the exponent value of the denominator

Mark needs to add heat to the solution in order to elevate the temperature from -4.3°C to a positive value

The amount of heat needed to increase a substance's temperature by one degree Celsius per gramme is known as its specific heat capacity. It is a characteristic of a substance that is intense and independent of the size or shape of the quantity under consideration. A substance's specific heat capacity is typically indicated by the letters "c" or "s"².

Mark needs to add heat to the solution in order to elevate the temperature from -4.3°C to a positive value.

. The bulk of the solution and its specific heat capacity affect the amount of heat needed.

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Answer:

The slope is -3

Step-by-step explanation:

Select two points on the line.  I have selected the points (1,5) and (2,2).  The slope is the change in y over the change in x.  The y values are 2 and 5,  The x values are 2 and 1.  You find the change by subtracting.

[tex]\frac{2-5}{2-1}[/tex] = [tex]\frac{-3}{1}[/tex] = -3

Another way to look at this is seeing the slope as the rise over the run.  If you start at (1,5) and only move right or left and up and down to get to (2,2), you would have to move straight down 3 spaces and then 1 space right.  Down is negative and right is positive, so the slope would be [tex]\frac{-3}{1}[/tex] which equals -3.

Helping in the name of Jesus.

The final equation of the circle is: (x - 4)²+(y - 3)² =8.

The set of all points in a plane that are equally spaced from a fixed point known as the centre is described by the equation of a circle.

It is usually written in the form (x-h)² + (y-k)² = r², where (h, k) represents the center and r represents the radius of the circle.

To find the equation of the circle, we need to find the center of the circle and its radius using the given endpoints of the diameter.

The circle's centre corresponds to the diameter PQ's midpoint.Taking the average of the x-coordinates and the average of the y-coordinates will yield the midpoint's coordinates:

x-coordinate of midpoint = (6 + 2)/2 = 4

y-coordinate of midpoint = (5 + 1)/2 = 3

(4, 3) is the center of circle.

The radius of circle is half the distance between the endpoints of      diameter:

r = 1/2 × √((6 - 2)² + (5 - 1)²) = 1/2 × √(16 + 16) = 1/2 × √(2) = 2√(2)

Therefore, the equation of the circle with center (4, 3) and radius 2√(2) is:

(x - 4)² + (y - 3)² = (2×(2))²

Simplifying and expanding  right-hand side:

(x - 4)² + (y - 3)² = 8

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Now, we can evaluate the improper integral of f(x) from 1 to infinity:
integral^infinity_1 1/(2x + 8) dx. Since, the improper integral diverges, the series also diverges by the Integral Test. Therefore, the answer is b. diverges.

Your answer: b. diverges

To apply the Integral Test, we first need to confirm that the function f(x) = 1/(2x + 8) is continuous, positive, and decreasing on the interval [1, ∞). Since the function meets these conditions, we can apply the Integral Test.

Now, let's evaluate the integral:
∫[1,∞] (1/(2x + 8)) dx

To solve this, we can use the substitution method:

let u = 2x + 8, so du = 2 dx. Now, when x = 1, u = 10, and when x → ∞, u → ∞.
Now, the integral becomes:
(1/2) ∫[10,∞] (1/u) du

This is an improper integral, and its form is a p-series where p = 1. We know that a p-series converges if p > 1 and diverges if p ≤ 1. In this case, p = 1, so the integral diverges.
Since the integral diverges, by the Integral Test, the given series also diverges.

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The answer is (a) [tex]$-\frac{9}{2}$[/tex].

To find the area of the bounded region determined by the curves [tex]$x^2=8y[/tex]and x + 4y - 4 = 0, we first need to find the intersection points of the two curves.

From the equation [tex]$x^2=8y$[/tex], we get [tex]$y=\frac{x^2}{8}$[/tex] Substituting this in the equation x + 4y - 4 = 0, we get [tex]$x+4\left(\frac{x^2}{8}\right)-4=0$[/tex], which simplifies to [tex]$x^2+8x-32=0$[/tex]. Solving for x, we get [tex]$x=-4\pm 4\sqrt{3}$[/tex].

Since the parabola [tex]$x^2=8y$[/tex] opens upwards, the area of the bounded region can be calculated as follows:

[tex]Area }=\int_{-4-4 \sqrt{3}}^{4 \sqrt{2}} \int_{\frac{x^2}{8}}^{(4-x) / 4} d y d x[/tex]

Integrating with respect to y first, we get:

[tex]\text { Area }=\int_{-4-4 \sqrt{3}}^{4 \sqrt{2}}\left(\frac{4-x}{4}-\frac{x^2}{8}\right) d x[/tex]

Simplifying and evaluating the integral, we get:

[tex]\text { Area }=\frac{9}{2}+\frac{16 \sqrt{3}}{3}-2 \sqrt{2}[/tex]

Therefore, the answer is (a)[tex]$-\frac{9}{2}$[/tex].

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answer: 9.1

The points are plotted at (-4,6) and (5,5)

i have attatched a photo with the values inputted into the distance formula!

Here, the data can be applied to understand the customer behavior, Stocked Pantry and preferences, adjust inventory, and optimize staffing and checkout procedures.

The Loaded Storage room supermarket places information about the quantity of things in stock by every client who goes through the express line.This information can be used in different ways to work on the store's tasks.

For example , the chief can use it to investigate client conduct and inclinations, distinguish well known things, and change stock likewise. The information can likewise be applied to enhance the store's staffing and checkout strategies.

In the event that the data shows that there is a top in express line traffic during specific times, the chief can plan gradually more staff during those times to guarantee speedy and productive help.

Generally, approaching this information can give significant experiences into the store's activities and assist the supervisor with pursuing informed choices that can further develop consumer loyalty and benefit.

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The complete question is

The manager at The Stocked Pantry grocery store can run a report to see the number of items purchased by each customer who goes through the express line. Customers in this line are allowed to purchase from 1 to 5 items. The table below shows the results from this morning.