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, [tex]V = x^{2} y[/tex]
Cost of top = $4 * [tex]x^{2}[/tex]
Cost of the bottom = $9 * [tex]x^{2}[/tex]
Cost of the sides = $3 * 4xy
Total cost = 4x² + 9x² + 12xy
Total cost = 13x² + 12xy
Total cost = $193
193 = 13x² + 12xy
[tex]y = \frac{193 - 13x^{2} }{12x}[/tex]
But volume, V = x²y
V = [tex]\frac{x^{2} (193 - 13x^{2}) }{12x}[/tex]
V = [tex]\frac{x (193 - 13x^{2}) }{12}[/tex]...................(1)
At maximum value, V' = 0
0 = [tex]\frac{193 - 39x^{2} }{12}[/tex]
[tex]39x^{2} = 193\\x^{2} = \frac{193}{39}[/tex]
x = 2.23
Put the value of x into (1)
[tex]V = \frac{2.23 (193 - 13*2.23^{2}) }{12}[/tex]
V = 23.85 ft³
