Question

An open box is to be made from a 5 ft by 9 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume that the box can have.

Answer 1

Answer: 21 ft³

Explanation:

Let x represent the height of the box.

Then the 5 ft width of cardboard is (5 - 2x) when creating the box

and the 9 ft length of cardboard is (9 - 2x) when creating the box.

Volume = length x width x height

= (9 - 2x)(5 - 2x)(x)

= 45x - 28x² + 4x³

Using Calculus to solve for x, set the derivative equal to zero and use the quadratic formula to solve for x:

V' = 45 - 56x + 12x²

0 = 12x² - 56x + 45

x = 3.6, x = 1.0

Use those values to find the width, length, and volume:

height(x) × width (5 - 2x) × length (9 - 2x) = Volume

3.63 × -2.26*

1 × 3 × 7 = 21

*width cannot be negative so the height cannot be 3.63