Answer :
Answer:
Explanation:
First we express teh other sides as expression of b (wide)
being the volume: a x b x c
and the base: a x b
if the base is long as wide then:
a = 2b
so we can construct:
2b x b x c = 2b^2 x c = 108
and we clear c:
c = 108/ (2b^2) = 54/b^2
Now we construct the cost formula:
ab x 0.50 (cost for the top part)
now the cost of the sides (there are four) and the bottom part:
(2bc + 2ac + ab) x 0.25
total cost: ab x 0.50 + (2bc + 2ac + ab) x 0.25
we replace a and c as expression of b:
[tex](2b)b \times 0.50 + (2b\frac{54}{b^2} + 2(2b)\frac{54}{b^2} + (2b)b) \times 0.25[/tex]
and now we solve for b:
[tex]b^2 + (\frac{108}{b} + \frac{216}{b} + 2b^2) \times 0.25[/tex]
[tex]b^2 + \frac{81}{b} + \frac{b^2}{2}[/tex]
[tex](1.5)b^2 + \frac{81}{b} [/tex]
Now we derivate the cost function: (considering)
[tex]a^{x} = xa\\\frac{1}{a^{x}} = -\frac{1x}{a^{x+1}}[/tex]
3b - 81/b^2 = 0
3b^3 = 81
b^3 = 81/3 = 27
b = third root of 27 = 3
now we solve for a:
a = 2b = 2(3) = 6
and last for c:
54/b^2 = 54/(3)^2 = 54/9 = 6
we check if we match the cubic inches:
a x b x c = 108
6 x 3 x 6 = 108