A rectangular box with a square base is to be constructed from material that costs $9 per ft2 for the bottom, $4 per ft2 for the top, and $3 per ft2 for the sides. Find the box of greatest volume that can be constructed for $193. Round your answer to 2 decimals.

V = 23.85 ft³

Step-by-step explanation:

The dimension of the square base = x ft by x ft

Height of box = y

Volume of the box,

Cost of top = $4 *

Cost of the bottom =  $9 *

Cost of the sides = $3 * 4xy

Total cost = 4x² + 9x² + 12xy

Total cost = 13x² + 12xy

Total cost = $193

193 = 13x² + 12xy

But volume, V = x²y

V =

V = ...................(1)

At maximum value, V' = 0

0 =

x = 2.23

Put the value of x into (1)

V = 23.85 ft³