Question 1

Solve the problem.
-A farmer's silo is the shape of a cylinder with a hemisphere as the roof. If the radius of the hemisphere is 10 feet and the height of the silo is h feet, express the volume of the silo as a function of h.

Accepted Answer

The volume of the silo is the sum of the volume of the cylinder and the volume of the hemisphere. The volume of the cylinder is $V_{cylinder} = \pi r^2 h = \pi (10)^2 h = 100\pi h$. The volume of the hemisphere is $V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (10)^3 = \frac{2000}{3} \pi$. Therefore, the total volume of the silo is $V(h) = 100\pi h + \frac{2000}{3} \pi$, but since the height of the cylinder part is $h - 10$ (not the total height $h$), the correct expression is $V(h) = 100\pi (h - 10) + \frac{2000}{3} \pi$.

Question 2

Solve the problem.
-A rectangular sign is being designed so that the length of its base, in feet, is 12 feet less than 4 times the height, h. Express the area of the sign as a function of h.

Accepted Answer

The area of a rectangle is given by the formula $A = \text{length} \times \text{height}$. According to the problem, the length of the base is $4h - 12$, where $h$ is the height. Therefore, the area as a function of $h$ is $A(h) = (4h - 12)h = 4h^2 - 12h$, which matches option D.

Question 3

Solve the problem.
-A wire 20 feet long is to be cut into two pieces. One piece will be shaped as a square and the other piece will be shaped as an equilateral triangle. Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle. What is the domain of A?

Accepted Answer

The answer of Solve the problem.
-A wire 20 feet long...

Question 4

Solve the problem.
-A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 11 inches by 23 inches by cutting out equal squares of side x at each corner and then folding up the sides as in the figure.
Express the volume V of the box as a function of x.

Accepted Answer

The answer of Solve the problem.
-A box with an open...

Question 5

Solve the problem.
-A farmer's silo is the shape of a cylinder with a hemisphere as the roof. If the height of the silo is 118 feet and the radius of the hemisphere is r feet, express the volume of the silo as a function of r.

Accepted Answer

The answer of Solve the problem.
-A farmer's silo is the...

Question 6

Solve the problem.
-The volume V of a square-based pyramid with base sides s and height h is V = 13 s2h. If the height is half of the
length of a base side, express the volume V as a function of s.

Accepted Answer

The answer of Solve the problem.
-The volume V of a...

Question 7

Solve the problem.
-Sue wants to put a rectangular garden on her property using 70 meters of fencing. There is a river that runs through her property so she decides to increase the size of the garden by using the river as one side of the
Rectangle. (Fencing is then needed only on the other three sides.) Let x represent the length of the side of the
Rectangle along the river. Express the garden's area as a function of x.

Accepted Answer

The total fencing available is 70 meters, and since the river acts as one side of the rectangle, only three sides need fencing. If x is the length of the side along the river, the remaining fencing, 70 meters, will be used for the other two sides and the side opposite the river. Let y be the length of the other two sides, so we have $2y + x = 70$, which simplifies to $y = \frac{70 - x}{2}$. The area of the rectangle, $A(x)$, is given by $x \times y$, which becomes $x \times \frac{70 - x}{2} = 35x - \frac{1}{2}x^2$.

Question 8

Solve the problem.
-The figure shown here shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 6 units long. Express the area A of the rectangle in terms of x.

Accepted Answer

The answer of Solve the problem.
-The figure shown here shows...

Question 9

Solve the problem.
-The price p and the quantity x sold of a certain product obey the demand equation: $$p = - \frac { 1 } { 5 } x + 500 , \{ x \mid 0 \leq x \leq 500 \}$$ What is the revenue to the nearest dollar when 400 units are sold?

Accepted Answer

Revenue is calculated as the product of price and quantity sold. Substituting $x = 400$ into the demand equation gives $p = -\frac{1}{5}(400) + 500 = -80 + 500 = 420$. Therefore, revenue for 400 units is $400 \times 420 = 168,000$.

Question 10

Solve the problem.
-A wire of length 8x is bent into the shape of a square. Express the area A of the square as a function of x.

Accepted Answer

The perimeter of the square is 8x, so each side is 2x. The area is side squared, so $A(x) = (2x)^2 = 4x^2$.

Question 11

Solve the problem.
-Let P = (x, y) be a point on the graph of y $$y = \sqrt { x }$$ . Express the distance d from P to the point (1, 0) as a function of x.

Accepted Answer

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Here, $x_1 = x$, $y_1 = \sqrt{x}$, $x_2 = 1$, and $y_2 = 0$. Substituting these values into the distance formula gives $d(x) = \sqrt{(1 - x)^2 + (0 - \sqrt{x})^2} = \sqrt{x^2 - 2x + 1 + x} = \sqrt{x^2 - x + 1}$.

Question 12

Solve the problem.
-A rectangular box with volume 268 cubic feet is built with a square base and top. The cost is $1.50 per square foot for the top and the bottom and $2.00 per square foot for the sides. Let x represent the length of a side of the
Base. Express the cost the box as a function of x.

Accepted Answer

The cost function is derived by calculating the cost of the top and bottom as $2 \times 1.5 \times x^2 = 3x^2$ and the cost of the sides as $4 \times 2 \times \frac{268}{x} = \frac{2144}{x}$. Thus, the total cost function is $C(x) = 3x^2 + \frac{2144}{x}$.

Question 13

Find the function.
-Find the function that is finally graphed after the following transformations are applied to the graph of y = x. The graph is shifted down 4 units, reflected about the y-axis, and finally shifted left 6 units.

Accepted Answer

The original function is y = x. The transformations are applied as follows: 1. Shifted down 4 units: y = x - 4.2. Reflected about the y-axis: y = -x - 4 (since reflection about the y-axis changes x to -x).3. Shifted left 6 units: This means adding 6 to the x-value, but since the function is already reflected, it effectively becomes y = -x - 6 - 4, which matches option B.

Question 14

Solve the problem.
-Elissa wants to set up a rectangular dog run in her backyard. She has 20 feet of fencing to work with and wants to use it all. If the dog run is to be x feet long, express the area of the dog run as a function of x.

Accepted Answer

The perimeter of the rectangle is given as $2x + 2y = 20$, where $x$ is the length and $y$ is the width. Solving for $y$, we get $y = 10 - x$. The area $A$ of the rectangle is $xy$, substituting $y$ with $10 - x$, we get $A(x) = x(10 - x) = 10x - x^2$.

Question 15

Solve the problem.
-Bob wants to fence in a rectangular garden in his yard. He has 82 feet of fencing to work with and wants to use it all. If the garden is to be x feet wide, express the area of the garden as a function of x.

Accepted Answer

The perimeter of the rectangle is given as $2x + 2y = 82$, where $x$ is the width and $y$ is the length. Solving for $y$, we get $y = 41 - x$. The area of the rectangle is $A = xy$, substituting $y$ gives $A(x) = x(41 - x) = 41x - x^2$.

Question 16

Solve the problem.
-From a 18-inch by 18-inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out. Express
The volume of the box as a function of x.

Accepted Answer

After cutting out squares of side length x from each corner, the dimensions of the box will be (18-2x) by (18-2x) by x. The volume, V(x), is therefore $V(x) = x(18-2x)(18-2x) = x(324-72x+4x^2) = 4x^3 - 72x^2 + 324x$.

Question 17

Solve the problem.
-The price $p$ and $x$, the quantity of a certain product sold, obey the demand equation
$p = - \frac { 1 } { 10 } x + 100 , \{ x \mid 0 \leq x \leq 1000 \}$ 
a) Express the revenue R as a function of x.
b) What is the revenue if 450 units are sold?
c) Graph the revenue function using a graphing utility.
d) What quantity x maximizes revenue? What is the maximum revenue?
e) What price should the company charge to maximize revenue?

Accepted Answer

The answer of Solve the problem.
-The price $p$ and $x$,...

Question 18

Solve the problem.
-A right triangle has one vertex on the graph of $$y = x ^ { 2 } \text { at } ( x , y ),$$ another at the origin, and the third on the
(positive) y-axis at (0, y). Express the area A of the triangle as a function of x.

Accepted Answer

The answer of Solve the problem.
-A right triangle has one...

Question 19

Solve the problem.
-A farmer has 800 yards of fencing to enclose a rectangular garden. Express the area A of the rectangle as a function of the width x of the rectangle. What is the domain of A?

Accepted Answer

The total perimeter is 800 yards, so if x is the width, the length would be $400 - x$ (since $2x + 2(400 - x) = 800$). The area $A(x)$ is $x(400 - x) = 400x - x^2$. The domain is $0 < x < 400$ because the width must be less than half the total perimeter for a rectangle to be formed.

Question 20

Solve the problem.
-A rectangle that is x feet wide is inscribed in a circle of radius 16 feet. Express the area of the rectangle as a function of x.

Accepted Answer

The diagonal of the rectangle is the diameter of the circle, which is 32 feet. Using the Pythagorean theorem, the length of the rectangle can be expressed as $\sqrt{1024 - x^2}$. The area is then $x \times \sqrt{1024 - x^2}$.

Quiz 1: Functions and Their Graphs

Solve the problem. -A farmer's silo is the shape of a cylinder with a hemisphere as the roof. If the radius of the hemisphere is 10 feet and the height of the silo is h feet, express the volume of the silo as a function of h.

Solve the problem. -A rectangular sign is being designed so that the length of its base, in feet, is 12 feet less than 4 times the height, h. Express the area of the sign as a function of h.

Solve the problem. -A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 11 inches by 23 inches by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.

Solve the problem. -A farmer's silo is the shape of a cylinder with a hemisphere as the roof. If the height of the silo is 118 feet and the radius of the hemisphere is r feet, express the volume of the silo as a function of r.

Solve the problem. -The volume V of a square-based pyramid with base sides s and height h is V = 13 s2h. If the height is half of the length of a base side, express the volume V as a function of s.

Solve the problem. -The price p and the quantity x sold of a certain product obey the demand equation: $p = - \frac { 1 } { 5 } x + 500 , \{ x \mid 0 \leq x \leq 500 \}$ What is the revenue to the nearest dollar when 400 units are sold?

Solve the problem. -A wire of length 8x is bent into the shape of a square. Express the area A of the square as a function of x.

Solve the problem. -Let P = (x, y) be a point on the graph of y $y = \sqrt { x }$ . Express the distance d from P to the point (1, 0) as a function of x.

Solve the problem. -A rectangular box with volume 268 cubic feet is built with a square base and top. The cost is $1.50 per square foot for the top and the bottom and $2.00 per square foot for the sides. Let x represent the length of a side of the Base. Express the cost the box as a function of x.

Find the function. -Find the function that is finally graphed after the following transformations are applied to the graph of y = x. The graph is shifted down 4 units, reflected about the y-axis, and finally shifted left 6 units.

Solve the problem. -Elissa wants to set up a rectangular dog run in her backyard. She has 20 feet of fencing to work with and wants to use it all. If the dog run is to be x feet long, express the area of the dog run as a function of x.

Solve the problem. -Bob wants to fence in a rectangular garden in his yard. He has 82 feet of fencing to work with and wants to use it all. If the garden is to be x feet wide, express the area of the garden as a function of x.

Solve the problem. -From a 18-inch by 18-inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out. Express The volume of the box as a function of x.

Solve the problem. -A right triangle has one vertex on the graph of $y = x ^ { 2 } \text { at } ( x , y ),$ another at the origin, and the third on the (positive) y-axis at (0, y). Express the area A of the triangle as a function of x.

Solve the problem. -A farmer has 800 yards of fencing to enclose a rectangular garden. Express the area A of the rectangle as a function of the width x of the rectangle. What is the domain of A?

Solve the problem. -A rectangle that is x feet wide is inscribed in a circle of radius 16 feet. Express the area of the rectangle as a function of x.