An engineer needs a metal box to shield sensitive electronic devices from external electric fields. One side of the box should be open so that it can be placed over the components. The box can be made from a 6 ft by 4 ft sheet of metal by cutting squares from the corners and folding up the sides.

a. What is the maximum volume of the box?

b. What dimensions of the box result in the maximum volume?

a. 8.4 ft³

b. 4.44 ft * 2.44 ft * 0.78 ft

Step-by-step explanation:

a. Maximum volume

We are creating a box with dimensions

l = 6 - 2x

w = 4 - 2x

h = x

V = lwh = x (6 - 2x) (4 - 2x)

We must determine the value of x that makes V a maximum.

One way is to plot the function V = x (6 - 2x) (4 - 2x).

The maximum appears to be at about (0.78, 8.4).

Thus, the maximum volume is 8.4 ft³.

b. Dimensions

l = 6 - 2 * 0.78 = 6 - 1.56 = 4.44 ft

w = 4 - 2 * 0.78 = 4 - 1.56 = 2.44 ft

h = 0.78 ft

The box with maximum volume has dimensions 4.44 ft * 2.44 ft * 0.78 ft.