1.A box with a square base andopen topmust have a volume of 37044cm^3. We wish to find the dimensions of the box that minimize the amount of material used.

First, find a formula for the surface area of the box in terms of onlyx, the length of one side of the square base.

[Hint: use the volume formula to express the height of the box in terms ofx]

Simplify your formula as much as possible.

A(x)=

Next, find the derivative, A'(x)

A'(x)=

Now, calculate when the derivative equals zero, that is, when A'(x)=0. [Hint: multiply both sides by x^2.]

A'(x)=0 when x=

we next have to make sure that this value of x gives a minimum value for the surface area.

Let's use the second derivative test. Find A''(x).

A''(x)=

Evaluate A''(x) at the x-value you gave above.

2.A rectangle inscribed with its base on the x-asis and its upper corners on the parabola y=4-x^2. what are the dimensions of such a rectangle with the greatest possible area?

Width=

Height=

3.A cylinder shaped can needs to be constructed to hold 600 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.05 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

Volume of a cylinder: V=r^2h

Area of the sides: A=2rh

Area of the top/bottom: A=r^2

To minimize the cost of the can:

Radius of the can:

Height of the can:

Minimum cost: cents