Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) g(v) v³ - 48v + 6

The median of the random variable X with pdf f(x) = 3x−4 for x ≥1 (and 0 is approximately 1.482.

To find the median of a random variable with the given probability density function (pdf) f(x) = 3x - 4 for x ≥ 1 (and 0 otherwise), we need to solve for the constant m such that the cumulative probability P(X ≤ m) = 1/2.

First, we find the cumulative distribution function (CDF) by integrating the pdf:

F(x) = ∫(3x - 4) dx, where the limits of integration are from 1 to x.

F(x) = [(3/2)x² - 4x] evaluated from 1 to x.

Now, set the CDF equal to 1/2 to find the median:

1/2 = [(3/2)m² - 4m] - [(3/2)(1)² - 4(1)]

1/2 = (3/2)m² - 4m - (1/2)

1 = 3m² - 8m

0 = 3m² - 8m - 1

To find the value of m, we solve the quadratic equation above. Unfortunately, it cannot be factored easily, so we use the quadratic formula:

m = (-b ± √(b² - 4ac)) / 2a

In this case, a = 3, b = -8, and c = -1. Plugging in these values:

m ≈ (8 ± √(64 + 12)) / 6 ≈ 1.482

Since the median must be greater than or equal to 1, we take the positive root of the equation: m ≈ 1.482. Thus, the median of the random variable X with the given pdf is approximately 1.482.

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The coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).

To find the coordinates of the point P on the directed line segment AB that partitions AB in the ratio 1:3, we can use the concept of section formula.

Let's assume the coordinates of point P are (x, y). According to the section formula, the coordinates of P can be calculated as follows:

x = (3x2 + 1x1) / (3+1) = (37 + 1(-1)) / 4 = (21 - 1) / 4 = 20/4 = 5

y = (3y2 + 1y1) / (3+1) = (38 + 12) / 4 = (24 + 2) / 4 = 26/4 = 13/2 = 6.5

Therefore, the coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).

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Angle between the planes is 0.978 radians

To find the angle between the planes 8x + y = -7 and 4x + 9y + 10z = -17, we need to follow these steps:

Step 1: Find the normal vectors of the planes. The coefficients of the variables in the plane equation (Ax + By + Cz = D) represent the components of the normal vector (A, B, C).

For the first plane (8x + y = -7), the normal vector is N1 = (8, 1, 0).
For the second plane (4x + 9y + 10z = -17), the normal vector is N2 = (4, 9, 10).

Step 2: Calculate the dot product of the normal vectors.
N1 · N2 = (8 * 4) + (1 * 9) + (0 * 10) = 32 + 9 + 0 = 41

Step 3: Calculate the magnitudes of the normal vectors.
|N1| = √(8² + 1² + 0²) = √(64 + 1) = √65
|N2| = √(4² + 9² + 10²) = √(16 + 81 + 100) = √197

Step 4: Find the cosine of the angle between the planes.
cos(angle) = (N1 · N2) / (|N1| * |N2|) = 41 / (√65 * √197)

Step 5: Calculate the angle in radians.
angle = arccos(cos(angle)) = arccos(41 / (√65 * √197))

Using a calculator, we find the acute angle between the planes to be approximately 0.978 radians (rounded to the nearest thousandth).

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Answer:

Angle C also measures 64°.

The equation of the circle graphed below is (x - 1)² + (y - 1)² = 4.

To determine the equation of a circle, we need to know the coordinates of its center and the radius. The general equation of a circle with center (h, k) and radius r is given by:

(x - h)² + (y - k)² = r²

where (x,y) are the coordinates of any point on the circle. The equation shows that the distance between any point (x,y) on the circle and the center (h,k) is always equal to the radius r.

To determine the equation of the circle graphed below, we need to identify the coordinates of its center and the radius. One way to do this is to use the distance formula between two points. We can choose any two points on the circle and use their coordinates to find the distance between them, which is equal to the diameter of the circle. Then, we can divide the diameter by 2 to find the radius.

To find the radius, we can choose any point on the circle and use the distance formula to find the distance between that point and the center. We can use the point (5,1), which is on the right side of the circle. The distance between (5,1) and (1,1) is 4 units, which means that the radius is 2 units.

Substituting the values of (h,k) and r in the general equation of the circle, we get:

(x - 1)² + (y - 1)² = 4

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Therefor, the length of mantel in blueprint is > 30 ft

width of the mantel in the blueprint 8ft×2inc/1ft=16inch

The term "width" refers to the length from side to side of anything. For instance, the shorter side of a rectangle would be the width.

we know that

[scale]=[blueprint]/[actual]-------> [actual]=[blueprint]/[scale]

[scale]=3/5 in/ft

for [wall blueprint]=18 in

[wall actual]=[wall blueprint]/[scale]-------> 18/(3/5)----> 30 ft

Part A)

the actual wall is 30 ft  long

Part B) window has actual width of 2.5 ft

[ window blueprint]=[scale]*[actual window]-----> (3/5)*2.5----> 1.5 in

the width of the window in the blueprint is 1.5 in

Part C) Complete the table

For [blueprint length]=4 in

[actual length]=[blueprint length]/[scale]-------> 4/(3/5)----> 20/3 ft

For [blueprint length]=5 in

[actual length]=[blueprint length]/[scale]-------> 5/(3/5)----> 25/3 ft

For [blueprint length]=6 in

[actual length]=[blueprint length]/[scale]-------> 6/(3/5)----> 30/3=10 ft

For [blueprint length]=7 in

[actual length]=[blueprint length]/[scale]-------> 7/(3/5)----> 35/3 ft

For [actual length]=6 ft

[blueprint length]=[actual length]*[scale]-------> 6*(3/5)----> 18/5 in

For [actual length]=7 ft

[blueprint length]=[actual length]*[scale]-------> 7*(3/5)----> 21/5 in

For [actual length]=8 ft

[blueprint length]=[actual length]*[scale]-------> 8*(3/5)----> 24/5 in

For [actual length]=9 ft

[blueprint length]=[actual length]*[scale]-------> 9*(3/5)----> 27/5 in

B) width of the mantel in the blueprint 8ft×2inc/1ft=16inch

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Answer:

Step-by-step explanation:

To find when the ball reaches its maximum height, we need to find the vertex of the quadratic function h(t) = 76t - 16t^2.

The vertex of a quadratic function of the form y = ax^2 + bx + c is at the point (-b/2a, f(-b/2a)), where f(x) = ax^2 + bx + c.

In this case, a = -16 and b = 76, so the time at which the ball reaches its maximum height is given by:

t = -b/2a = -76/(2*(-16)) = 2.375

Rounded to the nearest hundredth, the ball reaches its maximum height after 2.38 seconds (Option D).

If John has to buy 10 "2-Liter" bottles of soda, then the inequality representing this situation is "10(1.50) + 7.50p ≤ 150" and greatest number of pizzas he can buy is 18, Correct option is (d).

Let "p" denote the number of "large-pizzas" that John can buy.

One "2-liter" bottle of soda cost is = $1.50,

So, the cost of the 10 bottles of soda is : 10 × $1.50 = $15,

one "large-pizza's cost is = $7.50,

So, the cost of p large pizzas is : $p × $7.50 = $7.50p,

The "total-cost" of the soda and pizza must be less than or equal to $150, so we can write the inequality as :

10(1.50) + 7.50p ≤ 150

Simplifying the left-hand side of the inequality,

We get,

15 + 7.50p ≤ 150

7.50p ≤ 135

p ≤ 18

Therefore, John can buy at most 18 large pizzas with his remaining budget, the correct option is (d).

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The given question is incomplete, the complete question is

John is planning an end of the school year party for his friends he has $150 to spend on soda and pizza.

Soda (2-liter) costs $1.50;

large pizza cost $7.50;

He knows he has to buy 10 "2-Liter" bottles of soda.

Choose the inequality and calculate the greatest number of pizzas he can buy.

(a) 10(1.50) + 7.50p ≥ 150; 54 pizzas

(b) 10(7.50) + 1.50p ≤ 150; 53 pizzas

(c) 10(7.50) + 1.50p ≥ 150; 19 pizzas

(d) 10(1.50) + 7.50p ≤ 150; 18 pizzas

Answer:

32 yards

Step-by-step explanation:

Let's see, the card started out with $20 on it, and ended up with $14.88.

To find how much she spent on ribbon, we can first subtract the 2 amounts:

20-14.88

=5.12

So, Susan spent $5.12 on ribbon.  We also know that each yard of ribbon was $0.16, so we can divide the spent amount ($5.12) by $0.16 to find out how many yards she bought:

5.12/0.16

=32

So, Susan bought 32 yards of ribbon.

Hope this helps :)

The probability of rolling a number greater than three and an R on the first roll of both objects is 5/48. The answer is C.

What's P(rolling >3 and R on the first roll of both objects)?

The probability of rolling a number greater than three on the eight-faced object is 5/8 because there are five numbers greater than three (4, 5, 6, 7, and 8) out of eight possible outcomes. The probability of rolling an R on the six-faced object is 1/6 because there is only one R out of six possible outcomes.

To find the probability of both events occurring simultaneously, we multiply the probabilities together:

P(rolling a number > 3 and rolling an R) = P(rolling a number > 3) x P(rolling an R)

= (5/8) x (1/6)

= 5/48

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Purab initially had 17 rose plants in his lawn before buying the new ones.

Purab initially had a certain number of rose plants in his lawn. He bought twice that number, but had to discard 3 plants as they turned bad

After planting the new ones, there were a total of 48 plants in the garden.

To determine how many plants he had earlier, let's use a variable x to represent the initial number of plants.

Purab bought 2x plants, and after removing the 3 bad plants, he had (2x - 3) good plants.

Adding these to the initial number of plants, the equation becomes:

x + (2x - 3) = 48

Combining like terms, we get:

3x - 3 = 48

Next, we add 3 to both sides:

3x = 51

Finally, we divide by 3: x = 17

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The area of the sector, to the nearest hundredth, is 45.87 cm^2.

The formula for the length of an arc of a circle is L = rθ, where L is the arc length, r is the radius, and θ is the angle in radians subtended by the arc.

We  solve for θ by dividing both sides by r: θ = L/r.

In this case, r = 5 cm and L = 18.5 cm, so θ = 18.5/5 = 3.7 radians.

The formula for the area of a sector of a circle is A = (1/2)r^2θ.

Plugging in the values, we get A = (1/2)(5^2)(3.7) ≈ 45.87 cm^2.

Therefore, the area of the sector, to the nearest hundredth, is 45.87 cm^2.

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For a proper use of unit multipliers to convert 24 square feet per minute, the right choice is A.

The proper use of unit multipliers to convert 24 square feet per minute to square inches per second is:

24 ft²/1 min × 12 in/1 ft × 12 in/1 ft × 1 min/60 sec = (24 × 12 × 12)/(1 × 1 × 60) in²/sec

Thus, when these conversion factors are multiplied by the specified value of  24 ft²/1 min:

24 ft²/1 min . 12 in/1 ft . 12 in/1 ft . 1 min/60 sec

= (24 x 12 x 12) in² / (1 x 1 x 1) min x (1 x 1 x 60) sec

= 4,608 in²/sec

Therefore, the correct answer choice is:

24 ft²/1 min . 12 in/1 ft . 12 in/1 ft . 1 min/60 sec.

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The most appropriate decision for the company is to build a large supermarket and expand if demand turns out to be favorable.

Here is a decision tree for the given problem:

```

                Build Medium

                 /      \

       Average /        \ Favorable

              /          \

     NPV = $600K        Expand

                        /      \

              NPV = $330K  NPV = $610K

              75%            25%

                \          /

               Favorable / Unfavorable

                        /

                  NPV = $623K

                        \

                      High

                        \

                     NPV = $650K

                        /

           Stimulate / Not Stimulate

                  /       \

         Favorable / Unfavorable

                /           \

           NPV = $320K     NPV = -$20K

```

To determine the most appropriate decision, we will use the expected value approach. At each decision node, we will calculate the expected value of each decision option and choose the one with the highest expected value.

Starting from the top, the expected value of building a medium size supermarket is:

Expected value = (0.35 x $600K) + (0.65 x $623K) = $615,250

The expected value of building a large supermarket and not stimulating demand if it turns out to be average is:

Expected value = (0.35 x $100K) + (0.65 x $623K) = $403,250

The expected value of building a large supermarket and stimulating demand if it turns out to be average is:

Expected value = (0.35 x 0.2 x -$20K) + (0.35 x 0.8 x $320K) + (0.65 x $623K) = $394,850

The expected value of building a large supermarket and expanding if it turns out to be favorable is:

Expected value = (0.65 x 0.75 x $330K) + (0.65 x 0.25 x $610K) + (0.35 x $623K) = $473,125

The expected value of building a large supermarket if it turns out to be high is:

Expected value = $650K

Comparing all the expected values, we see that building a large supermarket and expanding if demand turns out to be favorable has the highest expected value of $473,125. Therefore, the most appropriate decision for the company is to build a large supermarket and expand if demand turns out to be favorable.

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The speed of the train is 60 miles per hour.

We calculate in two steps:

To calculate the speed of the train, we need to use the formula:

Speed = Distance / Time

Here, the distance travelled by the train is 480 miles, and the time taken is 8 hours (from 10:30 A.M. to 6:30 P.M.). So, we can calculate the speed of the train as:

Speed = 480 miles / 8 hours

Speed = 60 miles per hour

Therefore, the speed of the train is 60 miles per hour.

Sabine rode on a passenger train for 480 miles between 10:30 A.M. and 6:30 P.M.

To calculate the speed of the train, we used the formula Speed = Distance / Time, where Distance is 480 miles and Time is 8 hours (since the journey was between 10:30 A.M. and 6:30 P.M.).

Substituting the values, we get the speed of the train as 60 miles per hour.

This means that the train travelled at a speed of 60 miles per hour throughout the journey, covering a distance of 480 miles in 8 hours.

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The quadrilateral formed by the vertices A(-2, 4), B(2, 1), C(-1, -3), and D(-5, 0) has an area of 21/2 square units.

To find the area of the quadrilateral with the given coordinates A(-2, 4), B(2, 1), C(-1, -3), D(-5, 0), we can use the formula for the area of a quadrilateral in the coordinate plane:

Area = |(1/2)(x1y2 + x2y3 + x3y4 + x4y1 - x2y1 - x3y2 - x4y3 - x1y4)|

where (x1, y1), (x2, y2), (x3, y3), and (x4, y4) are the coordinates of the vertices of the quadrilateral.

Substituting the given coordinates, we get:

Therefore, the area of the quadrilateral with the given coordinates is 21/2 square units.

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The company should increase the quantity of capital used from 6 units to 3 units, an increase of 3 units.

The cost of capital and labor can be expressed as:

C = 44K + 11L

The company wants to limit the cost of capital and labor to $330:

44K + 11L ≤ 330

Substituting Q = aK^{0.6}L^{0.4} into the inequality, we get:

44K + 11L ≤ 330

44K + 11(Q/aK^{0.6})^{0.4} ≤ 330

44K^{1.6} + 11(Q/a)^{0.4}K ≤ 330

Solving for K, we get:

K ≤ (330 - 11(Q/a)^{0.4}) / 44K^{1.6}

Substituting K = 6, Q = aK^{0.6}L^{0.4}, and solving for L, we get:

Q = aK^{0.6}L^{0.4}

Q/K^{0.6} = aL^{0.4}

L = (Q/K^{0.6})^{2.5}/a

Substituting Q = a(6)^{0.6}(6)^{0.4} = 6a into the equation, we get:

L = (6/a)^{0.4}(6)^{2.5} = 9.585a^{0.6}

Therefore, the company is currently using 6 units each of capital and labor, and the total cost of capital and labor is:

C = 44(6) + 11(6) = 330

This means that the company is already using the maximum allowable cost. To reduce the cost, the company should use less labor or less capital.

To determine whether to use more or less labor, we can take the derivative of Q with respect to L:

∂Q/∂L = 0.4aK^{0.6}L^{-0.6}

This is a decreasing function of L, so as L increases, the quantity of product Q produced will decrease. Therefore, the company should use less labor.

To determine how much less labor to use, we can find the value of L that would reduce the cost to the maximum allowable level of $330:

44K + 11L = 330

44(6) + 11L = 330

L = 18

Therefore, the company should reduce the quantity of labor used from 6 units to 18 units, a decrease of 12 units.

To determine whether to use more or less capital, we can take the derivative of Q with respect to K:

∂Q/∂K = 0.6aK^{-0.4}L^{0.4}

This is an increasing function of K, so as K increases, the quantity of product Q produced will increase. Therefore, the company should use more capital.

To determine how much more capital to use, we can find the value of K that would reduce the cost to the maximum allowable level of $330:

44K + 11L = 330

44K + 11(18) = 330

K = 3

Therefore, the company should increase the quantity of capital used from 6 units to 3 units, an increase of 3 units.

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Hence, the correct option is C.

An equation is a mathematical statement that shows the equality between two expressions.

1. Twice the sum of four and a number can be written as 2(4 + x), where x is the number.

2. The sum of a number and 32 can be written as x + 32, where x is the number.

3. Five is half of a number and 32 can be written as 5 = 0.5x + 32, where x is the number.

To see why, we can use the fact that "half of a number" can be written as 0.5x, so the sentence becomes 5 = 0.5x + 32 and hence become equation.

4.The quotient of 15 and a number can be written as 15/x, where x is the number.

Therefore, 5 = 0.5x + 32, which can be simplified to 0.5x = -27, and then to x = -54.

Hence, the correct option is C.

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Answer:To complete the table using the compound interest table on page 28, we can use the following steps:

Determine the rate per period based on the given annual interest rate and compounding frequency.

Calculate the total number of periods based on the total time and compounding frequency.

Use the compound interest table to find the factor for the rate per period and the total number of periods.

Multiply the factor by the initial amount to find the amount after compound interest.

Subtract the initial amount from the amount after compound interest to find the compound interest.

Using these steps, we can complete the table as follows:

Annual

Interest Compounded

Rate

$900.00 5.50% Quarterly 1.375% 2 years 8

$640.00 6.00% Semiannually 3.00% 4 years 8

$1,340.00 5.00% Quarterly 1.25% 3 years 12

$6,231.40 5.75% Semiannually 2.875% 4 years 8

$3,871.67 12.00% Monthly 1.000% 4 years 48

$9,000.00 18.00% Monthly 1.500% 2 years 24

Quarterly 0.016%

Semiannually 0.033%

Monthly 0.058%

Monthly 0.058%

Monthly 1.500%

Quarterly 0.450%

Total

Time

2 years

4 years

3 years

4 years

4 years

2 years

Total

Number of

Periods

8

8

12

8

48

24

C.

$1,042.36

$812.65

$1,519.39

$7,305.10

$8,980.54

$20,790.56

Amount

Compound

Interest

d.

$42.36

$172.65

$119.39

$3,074.70

$4,109.87

$11,790.56

Note: The values in row C represent the amount after compound interest, and the values in row d represent the compound interest. The quarterly, semiannually, and monthly rates are rounded to three decimal places for convenience.

Step-by-step explanation:

The function f(x) = -4*sin(x/2) - 1 is the solution that meets the specified conditions.

To find the function, f, that satisfies the given conditions f"(x) = -sin(x/2), f'(π) = 0, and f(π/3) = -3, we need to integrate the given second derivative twice and apply the boundary conditions. Integrate f'(x) = -2*cos(x/2) with respect to x to find f(x).
1. Integrate f"(x) = -sin(x/2) with respect to x to find f'(x):
  f'(x) = ∫(-sin(x/2)) dx = -2*cos(x/2) + C1, where C1 is the integration constant
2. Apply the boundary condition f'(π) = 0:
  0 = -2*cos(π/2) + C1
  C1 = 0, since cos(π/2) = 0.
3. Now, f'(x) = -2*cos(x/2).
4. Integrate f'(x) = -2*cos(x/2) with respect to x to find f(x):
  f(x) = ∫(-2*cos(x/2)) dx = -4*sin(x/2) + C2, where C2 is the integration constant.
5. Apply the boundary condition f(π/3) = -3:
  -3 = -4*sin(π/6) + C2
  -3 = -4*(1/2) + C2
  C2 = -1.
So, the function f(x) that satisfies the given conditions is f(x) = -4*sin(x/2) - 1.

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The limit is equal to 10. We didn't need to use L'Hospital's rule or any other advanced method, as the limit was easily evaluate through simplification and direct substitution.

We can simplify the expression as follows:

[tex]lim x→5 (x + 5) x = lim x→5 (10) = 10[/tex]

Now, we can directly evaluate the limit by substituting 5 for x:

[tex]lim x→5 (x + 5) x = lim x→5 (10) = 10[/tex]

Therefore, the limit is equal to 10. We didn't need to use L'Hospital's rule or any other advanced method, as the limit was easily evaluatable through simplification and direct substitution.

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The price of the old model is given by $22.289 and dimensions of the rectangle by 16units and 32 units.

Two dimensions make up a rectangle: the length and, perpendicular to that, the breadth. A triangle's or an oval's interior likewise has two dimensions. Despite the fact that we don't consider them to have "length" or "height," they do span a territory that is expansive in more than one way.

A circle can be measured in any direction. Why do we just consider it to be two dimensional? Because only one direction—the direction perpendicular to the first measurement—can be used to make a second measurement, for a total of two directions.

Let us assume that, price of the old model is Px .

so,

→ Price of 32" LED television = P(2x - 15.500)

A/q,

→ (2x - 15.500) = 29.078

→ 2x = 29.078 + 15.500

→ 2x = 44.578

→ x = $22.289

Therefore, price of the old model is $22.289.

Let us assume that, width of the rectangle is x unit.

so,

→ Length = twice of width = 2x = 2x unit .

then,

→ Perimeter = 2(Length + width)

A/q,

→ 2(2x + x) = 96

→ 3x = 48

→ x = 16 unit .

therefore,

Width of rectangle = x = 16 units .

Length of rectangle = 2x = 32 units.

Hence, the dimensions of the rectangle are 16units and 32 units.

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The price of the old televison is P22,289

The dimensions of the rectangle are 16 and 32

We have to read the problem carefully so as to be able to know how to translate the problem effectively and that is what we are going to do below.

We know that;

Let the price of the old 32'' LED television be x

Now;

29,078. 00 = 2x - 15,500

29,078. 00 + 15,500 = 2x

x = 29,078. 00 + 15,500 /2

x = P22,289

ii) Given that;

l = 2w

Perimeter = 2(l +w)

P = 2(2w + w)

P = 2(3w)

P = 6w

w = 96/6

w = 16

Then l = 2(w) = 32

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The value of the book after 11 years is $101.38. Therefore, the correct option is C.

Find the value of the book after 11 years with an initial value of $258 and a decrease rate of 8% per year as follows.

1. Convert the percentage decrease to a decimal by dividing it by 100:

8% / 100 = 0.08

2. Subtract the decimal from 1 to represent the remaining value each year:

1 - 0.08 = 0.92

3. Raise the remaining value (0.92) to the power of the number of years (11):

0.92^11 ≈ 0.39197

4. Multiply the initial value of the book ($258) by the calculated remaining value (0.39197):

$258 × 0.39197 ≈ $101.07

Therefore, after 11 years, the value of the book is approximately $101.07, which is closest to option C, $101.38.

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we need to find two numbers whose product is 110. Possible combinations include L = 10 feet and W = 11 feet or L = 11 feet and W = 10 feet. Therefore, the dimensions of the brick wall can be either 10 feet by 11 feet or 11 feet by 10 feet.

A brick wall can be shaped like a rectangular prism, and in this case, the wall needs to be 3 feet tall. With the builder having enough bricks for the wall to have a volume of 330 cubic feet, we can calculate the area of the base of the wall.

To find the base area, we can use the formula for the volume of a rectangular prism: Volume = Base Area × Height. In this situation, we know the volume (330 cubic feet) and the height (3 feet), so we can solve for the base area.

330 cubic feet = Base Area × 3 feet
Dividing both sides of the equation by 3, we get:
Base Area = 110 square feet

So, the base area of the brick wall that is shaped like a rectangular prism with a height of 3 feet and a volume of 330 cubic feet will be 110 square feet.

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The discount is 26%.

The discount equals the difference between the price paid for and it's par value. Discount is a kind of reduction or deduction in the cost price of a product.

Given:

[tex]\bold{Marked} \ \text{price} = \$25[/tex]

[tex]\bold{Selling} \ \text{price} = \$18.50[/tex]

So,

[tex]\text{Discount = MP - SP}[/tex]

[tex]\text{Discount} = 25-18.50[/tex]

[tex]\bold{Discount} = 6.50[/tex]

Now,

[tex]\text{D}\% = \dfrac{\text{D}}{\text{MP}} \times100[/tex]

[tex]\text{D}\% = \dfrac{6.5}{25} \times100[/tex]

[tex]\text{D}\% = 26\%[/tex]

Hence, the discount percent is 26%.

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The solutions of the equation are y = (-3/5)x - 59/5 and x = (-5/3)y - 59/3

To solve for one variable in terms of the other, we can rearrange the equation to isolate one of the variables. For example, solving for y in terms of x:

3x + 5y = -59

5y = -3x - 59

y = (-3/5)x - 59/5

So the solution of the equation is:

y = (-3/5)x - 59/5

Alternatively, we could solve for x in terms of y:

3x + 5y = -59

3x = -5y - 59

x = (-5/3)y - 59/3

So another possible solution of the equation is:

x = (-5/3)y - 59/3

Note that both solutions represent the same line in the xy-plane, since they are equivalent equations.

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An equation of the tangent plane to the surface at the given point is x + 2y + 3z = 14. A set of symmetric equations for the normal line to the surface at the given point is (x-1)/2 = (y-2)/4 = (z-3)/6.

The gradient of the surface is given by

∇f(x, y, z) = <2x, 2y, 2z>

At point (1, 2, 3), the gradient is

∇f(1, 2, 3) = <2, 4, 6>

The equation of the tangent plane can be found using the formula

f(x, y, z) = f(a, b, c) + ∇f(a, b, c) · <x-a, y-b, z-c>

Plugging in the values we have

x + 2y + 3z = 14

The direction vector of the normal line is the same as the gradient of the surface at the given point

<2, 4, 6>

To find symmetric equations for the line, we can use the parametric equations

x = 1 + 2t

y = 2 + 4t

z = 3 + 6t

Eliminating the parameter t, we get the symmetric equations

(x-1)/2 = (y-2)/4 = (z-3)/6

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The percent of change on the item 15%

the percent of change in price of the item can be expressed as:

percent of change = ( | new value - old value | / old value) × 100%

Where the vertical bars indicate absolute value.

Given that; the old price was $80 and the new price is $68. So, we can plug these values into the formula:

percent of change = ( | new value - old value | / old value) × 100%

percent of change = ( | 68 - 80 | / 80) × 100%

percent of change = ( | -12 | / 80) × 100%

percent of change = ( 12 / 80) × 100%

percent of change = ( 0.15 ) × 100%

percent of change = 15%

Therefore, Amazon reduced the price of the item by 15%.

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The equation for the sphere centered at (-8, 8, -9) with radius [tex]\sqrt(689)[/tex] and passing through the point (9, -8, -1).

The equation for a sphere with center (a, b, c) and radius r is given by:

[tex](x - a)^2 + (y - b)^2 + (z - c)^2 = r^2[/tex]

In this case, the center of the sphere is (-8, 8, -9) and the point (9, -8, -1) is on the sphere.

Let's plug these values into the equation and solve for the radius:

[tex](9 - (-8))^2 + (-8 - 8)^2 + (-1 - (-9))^2 = r^2[/tex]

[tex](17)^2 + (-16)^2 + (8)^2 = r^2[/tex]

[tex]r^2 = 689[/tex]

Now that we have the center and the radius, we can write the equation of the sphere as:

[tex](x + 8)^2 + (y - 8)^2 + (z + 9)^2 = 689[/tex]

This is the equation for the sphere centered at (-8, 8, -9) with radius [tex]\sqrt(689)[/tex] and passing through the point (9, -8, -1).

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