optimization problem help!

Question

An open box is to be made out of a 8-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Dimensions of the bottom of the box:           x               
Height of the box:              

Answer 1

Hi Brittany,

 

I'll help get you started.

 

After cutting out the squares and bending up the sides, the open box will have dimensions:

L=14-2x, W=8-2x, H=x. That's because cutting out the two squares reduces each side of the box by x inches twice.

 

So a formula for the volume of the resulting box is: V = (14-2x)(8-2x)x

 

V is a function of x. Find dV/dx, set the derivative equal to zero (like finding critical numbers for the function). The x value(s) that produces zero for the derivative are the candidates to produce the open box with the maximum volume. To be thorough technically you should do a test to prove that a maximum value of V is the result (as opposed to a minimum).