please help i have a quiz tommorow

Question

A box with a square base and open top must have a volume of 32000 cm^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 only x, the length of one side of the square base.

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

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` 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.

__________________

NOTE: Since your last answer is positive, this means that the graph of `A(x)` is concave up around that value, so the zero of `A'(x)` must indicate a local minimum for `A(x)`. (Your boss is happy now.)

Answer 1

the side of the base = 40 cm= x

the height = 20 cm = h

A = x^2 +4xh

A' = 2x - 128,000/x^2 = 0

multiply by x^2 to get

2x^3 - 128,000=0

x^3 = 64,000

x = 40 cm

h= 32,000/16,00 = 20 cm

It's always nice when it comes out to an integer