An open box is to be constructed so that the length of the base is 4 times large

Question

An open box is to be constructed so that the length of the base is 4 times larger than the width of the base. If the cost to construct the base is 5 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 = 65 cubic feet which would minimize the cost of construction. The values for the dimension of the base are: 4650/200) (13) x (650/200) (1/3) The height of the box is: 65K4 (650/200*2/3) Submit Answer Incorrect. Tries 4/8 Previous Tries

Explanation / Answer

L = 4w
Base osts 5 per sqft
The 4 walls costs 1 dollar per sqft

Now, volume = w*4w*h = 65

So, h = 65/(4w^2)

The base area = LW = w*4w = 4w^2

Cost of base = 4w^2 * 5 = 20w^2 dollars

The 4 walls have area = 2Lh + 2wh

A = 8wh + 2wh = 10wh

Cost of these 4 walls = 10wh * 1 = 10wh dollars

Total cost is :
20w^2 + 10wh

Now, plug in h = 65/(4w^2)

C = 20w^2 + 10w * (65/(4w^2))

C = 20w^2 + 162.5/w

Now, for min cost
dC/dw = 0 :

40w - 162.5/w^2 = 0

w^3 = 6500

So, w = 6500^(1/3)

Now, length is :
L = 4*(6500)^(1/3)

And height, h is :
h = 65/(4w^2)

h = 65/(4*(6500)^(2/3))

So, dimensions of base are :
4*(6500)^(1/3) ----> ANS
x
6500^(1/3) ----> ANS

Ht of box is :
65/(4*(6500)^(2/3)) ----> ANS

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