Question

have a 12 inch square piece of thin material and want to make an open box by cutting small squares from the corners of our material and folding the sides up. The question is, “Which cut produces the box of maximum volume?”​

Answer 1

Answer:

Step-by-step explanation:

his is a classic optimization problem in calculus. Let s be the length of the sides of the squares to be cut out.

Then the dimensions of the open box will be 20 - 2s by 20 -2s by s .

The volume is thus V = (20 - 2s)2 s

To find the value of s which maximizes this, compute the derivative dV/ds and set it equal to zero.

Using the product rule, power rule and chain rule dV/ds = (20 - 2s)2 - 4 s (20 - 2s)

Setting this equal to zero gives (20 - 2s) - 4 s = 0 with the solution

s = 10/3

The maximum volume is then (20 - 20/3)2 (10/3) = 16000/27