What dimensions minimize the cost?

Question

a rectangular box with a volume of 1216 feet ^3 is to be constructed with a square base and top. The cost per square foot for the bottom is 20 cents, for the top 10 cents, and for the sides is 2.5 cents. What dimensions minimize the cost?

Answer 1

Let h = the height of the box and a = the length of a side on the top & bottom.  The volume of the box is:

 

Volume = 1216 ft³ = a²h, so h = 1216/a²

 

The surface area of the top and bottom is a².  The surface area of the lateral sides is 4ah.

 

Cost = Cost of top + Cost of bottom + Cost of sides

 Cost = ($0.10)a² + ($0.20)a² + ($0.025)(4ah)

 

Substitute h=1216/a²:

 

Cost = (0.3)a² + (0.025)(4*a*1216/a²)

Cost = (0.3)a² + (121.6)/a

 

To find the minimum cost, take the first derivative of Cost wrt a, set it to zero, and solve for a:

 

d(cost)/da = (0.6)a - 121.6/a²

 

0 = (0.6)a - 121.6/a²

121.6/a² = (0.6)a

121.6/(0.6) = a³

(202.67)^1/3 = a

 

5.87 ≅ a

 

h = 1216/a² = 1216/(5.87)² = 35.29