A box with a square base and open top must have a volume of 296352 c m 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.

Step-by-step explanation:

Given a box with a square base and an open top which must have a volume of 296352 cubic centimetre. We want to minimize the amount of material used.

Step 1:

Let the side length of the base =x

Let the height of the box =h

Since the box has a square base

Volume,

Surface Area of the box = Base Area + Area of 4 sides

Step 2: Find the derivative of A(x)

Step 3: Set A'(x)=0 and solve for x

Step 4: Verify that x=84 is a minimum value

We use the second derivative test

Since the second derivative is positive at x=84, then it is a minimum point.

Recall:

Therefore, the dimensions that minimizes the box surface area are:

the surface area is 124cm²

step-by-step explanation:

length: 4cm

width: 9cm

height:   2cm