Question

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.

Answer 1

Answer:

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, $V = x^{2} y$

Cost of top = $4 * $x^{2}$

Cost of the bottom =  $9 * $x^{2}$

Cost of the sides = $3 * 4xy

Total cost = 4x² + 9x² + 12xy

Total cost = 13x² + 12xy

Total cost = $193

193 = 13x² + 12xy

$y = \frac{193 - 13x^{2} }{12x}$

But volume, V = x²y

V = $\frac{x^{2} (193 - 13x^{2}) }{12x}$

V = $\frac{x (193 - 13x^{2}) }{12}$...................(1)

At maximum value, V' = 0

0 = $\frac{193 - 39x^{2} }{12}$

$39x^{2} = 193\\x^{2} = \frac{193}{39}$

x = 2.23

Put the value of x into (1)

$V = \frac{2.23 (193 - 13*2.23^{2}) }{12}$

V = 23.85 ft³