Question

An open box is to be constructed so that the length of the base is 3 times larger than the width of the base. If the cost to construct the base is 4 dollars per square foot and the cost to construct the four sides is 1 dollars per square foot, determine the dimensions for a box to have volume

Answer 1

Answer:

Step-by-step explanation:

Let assume that the volume = 88 cubic feet

Then:

$L = 3w --- (1)volume = l \times w \times h \\ \\ 88 = (3w) \times w \times h \\ \\ 3w^2 h= 88 \\ \\ h = \dfrac{88}{3w^2}--- (2)$

The construction cost now is:

$C = 4(l \times w) + 2 ( w \times h) + 2(l \times h) \\ \\ C = 4(3w^2) + 2(w \times \dfrac{88}{3w^2}) + 2(3w \times \dfrac{88}{3w^2}) \\\\ C = 12w^2 + \dfrac{176}{3w} + \dfrac{176}{w}$

Now, to determine the minimum cost:

$\dfrac{dC}{dw}= 0 \\ \\ \implies 24 w - \dfrac{176}{3w^2}- \dfrac{176}{w^2}=0 \\ \\ 24 w ^3 = \dfrac{176}{w^2}(\dfrac{1}{3}+1) \\ \\ 24 w ^3 = \dfrac{176(4)}{3} \\ \\ w^3 = \dfrac{88}{3(3)}$

$w = \dfrac{88}{3(3)}^{1/3} \ feet$

Now;

$length = 3 ( \dfrac{88}{3(3)})^{1/3} \ feet$

$height = ( 88})^{1/3} (3)^{1/3} \ feet$