# 1. A cylindrical container must be constructed to contain 250 cubic inches of liquid. a. Express the entire surface area of…

1. A cylindrical container must be constructed to contain 250 cubic inches of liquid.

a. Express the entire surface area of the container as a function of its radius r and height h. Don’t forget the top and bottom.

b. Express the volume as a function of r and h, and set this expression equal to 250.

c. Use your work from part b to solve for h in terms of r. Then substitute this expression for h into your area expression from part a.

d. Using a graphing technique, find the value of r that makes the surface area a minimum.

e. What dimensions of the container should the manufacturer use if his goal is to minimize the amount of material used in its manufacture?

2. A rectangular box with a square base of length s and height y is to have a volume of 16 cubic feet. The cost of top and bottom material for the box is 25 cents per square foot, and the cost for the sides is 10 cents per square foot.

a. Find an expression for the volume of the box in terms of s and y, and set this equal to 16.

b. Find an expression for the cost of the material used to make the box in terms of s and y.

c. Use your work from part a to express the cost of the box in terms of s only.

d. Find the dimensions of the box that will minimize the cost of materials used to make it.

3. A box without a lid is formed by taking a piece of cardboard that is 40 inches by 20 inches, cutting out square pieces from the four corners, and then bending up the sides to form a box.

a. Find an expression for the volume of the box in terms of the side length x of the cut-out squares.

b. Find the value of x that yields maximum volume.

4. Consider the function . Use the graph of this function to answer the questions below.

a. What is the domain of this function?

b. What is its range?

c. On which interval(s) is the function increasing?

d. On which interval(s) is it decreasing?

e. Does the function have a horizontal asymptote? If so, what is it?

f. Identify any vertical asymptotes.

g. Does this function have an inverse on the interval (-2, 1)?

h. Does it have an inverse on (1, ∞)?

a. Express the entire surface area of the container as a function of its radius r and height h. Don’t forget the top and bottom.

b. Express the volume as a function of r and h, and set this expression equal to 250.

c. Use your work from part b to solve for h in terms of r. Then substitute this expression for h into your area expression from part a.

d. Using a graphing technique, find the value of r that makes the surface area a minimum.

e. What dimensions of the container should the manufacturer use if his goal is to minimize the amount of material used in its manufacture?

2. A rectangular box with a square base of length s and height y is to have a volume of 16 cubic feet. The cost of top and bottom material for the box is 25 cents per square foot, and the cost for the sides is 10 cents per square foot.

a. Find an expression for the volume of the box in terms of s and y, and set this equal to 16.

b. Find an expression for the cost of the material used to make the box in terms of s and y.

c. Use your work from part a to express the cost of the box in terms of s only.

d. Find the dimensions of the box that will minimize the cost of materials used to make it.

3. A box without a lid is formed by taking a piece of cardboard that is 40 inches by 20 inches, cutting out square pieces from the four corners, and then bending up the sides to form a box.

a. Find an expression for the volume of the box in terms of the side length x of the cut-out squares.

b. Find the value of x that yields maximum volume.

4. Consider the function . Use the graph of this function to answer the questions below.

a. What is the domain of this function?

b. What is its range?

c. On which interval(s) is the function increasing?

d. On which interval(s) is it decreasing?

e. Does the function have a horizontal asymptote? If so, what is it?

f. Identify any vertical asymptotes.

g. Does this function have an inverse on the interval (-2, 1)?

h. Does it have an inverse on (1, ∞)?

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