An open box is to be constructed so that the length of the base is 3 times large

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 2 dollars per square foot and the cost to construct the four sides is 3 dollars per square foot, determine the dimensions for a box to have volume = 50 cubic feet which would minimize the cost of construction.

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 2 dollars per square foot and the cost to construct the four sides is 3 dollars pe square foot, determine the dimensions for a box to have volume- 50 cubic feet which would minimize the cost of construction. The values for the dimension of the base are: The height of the box is: Submit Answer Tries 0/8

Explanation / Answer

L = 3w
w = w

V = 3w*w*h = 50

So, h = 50/(3w^2)

Now, the total surface area is :
LW + 2WH + 2LH

S = 3w^2 + 2wh + 2*3w*h

Now, cost of the 4 sides is : 3 dollars per sqft
So, cost of 4 sides = (2wh + 6wh)*3 = 24wh dollars

Cost of the bottom is : 2*3w^2 = 6w^2

So, total cost
C = 6w^2 + 24wh ---> to be minimized

Now, plug back for h :
C = 6w^2 + 24w(50/(3w^2))

Simplifyin' :
C = 6w^2 + 400/w

Derivin' :
C' = 12w - 400/w^2 = 0

w^3 = 400/12

w^3 = 100/3

So, w = (100/3)^(1/3) ---> ANS

So, length of base is : 3*(100/3)^(1/3) ---> ANS

Now using h = 50/(3w^2)
we get

h = 50/(3*(100/3)^(2/3))

h = 50/3 * (3/100)^(2/3) ---> ANS

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