Answer:
Step-by-step explanation:
Let assume that the volume = 88 cubic feet
Then:
[tex]L = 3w --- (1)volume = l \times w \times h \\ \\ 88 = (3w) \times w \times h \\ \\ 3w^2 h= 88 \\ \\ h = \dfrac{88}{3w^2}--- (2) \\ \\[/tex]
The construction cost now is:
[tex]C = 4(l \times w) + 2 ( w \times h) + 2(l \times h) \\ \\ C = 4(3w^2) + 2(w \times \dfrac{88}{3w^2}) + 2(3w \times \dfrac{88}{3w^2}) \\\\ C = 12w^2 + \dfrac{176}{3w} + \dfrac{176}{w}[/tex]
Now, to determine the minimum cost:
[tex]\dfrac{dC}{dw}= 0 \\ \\ \implies 24 w - \dfrac{176}{3w^2}- \dfrac{176}{w^2}=0 \\ \\ 24 w ^3 = \dfrac{176}{w^2}(\dfrac{1}{3}+1) \\ \\ 24 w ^3 = \dfrac{176(4)}{3} \\ \\ w^3 = \dfrac{88}{3(3)}[/tex]
[tex]w = \dfrac{88}{3(3)}^{1/3} \ feet[/tex]
Now;
[tex]length = 3 ( \dfrac{88}{3(3)})^{1/3} \ feet[/tex]
[tex]height = ( 88})^{1/3} (3)^{1/3} \ feet[/tex]
