Expert: Ahmed Salami - 12/29/2009

Question
I need to design a 500 ft^3 square based, open top tank. i need to find the dimensions for the base and the height that will make the tank weigh as little as possible. Assume that the weight is minimized when the total area of the bottom and the 4 sides is minimized.  

The answer is given, 10 ft by 10 ft by 5 ft. However, I need to show all of the work to get to this answer.

Thanks!

Answer
Hi Leanna,
If the tank has base length b and height h and should be 500ft³ in volume, then
Volume = Ah    (where A is the base area)
500 = b²h
h = 500/b²
The total surface area S of the box would be the base area plus the area of the four sides,
S = A + 4(bh)
 = b² + 4bh
 = b² + 4b(500/b²)
 = b² + 2000/b
Now, we need to find the values of b that minimizes S. To do this we find dS/db and equate to zero.
dS/db = 2b - 2000/b²
equating to zero,
2b - 2000/b² = 0
2b = 2000/b²
b³ = 1000
b = 10
and
h = 500/b²
 = 500/100
 = 5

Regards