Expert: Ahmed Salami - 12/6/2005

Question
The motorcycles memorabilia Company wishes to design an open top box with a square base whose volume is exaclty 32 cubic feet. What are the dimensions of the box requiring the least amount of materials?

Answer
Hi Fran,
Forgive the time taken.
The volume of the tank
V = lx^2
where l is the height of the tank and x is the length
of the square base.
But we require a specific volume of 32, so
32 = lx^2
l = 32/x^2
The amount of material to be used would depend on the
surface area of the tank. For a tank open at the top
the total surface area would be the base area(x^2)
+ the area of the four sides(lx each).
Therefore, surface area
A = x^2 + 4lx
replacing l by 32/x^2 gives us
A = x^2 + 4(32/x^2)x
 = x^2 + 128/x
The minimum area would occur when dA/dx = 0
dA/dx = 2x - 128/x^2
at dA/dx = 0
2x - 128/x^2 = 0
2x = 128/x^2
x^3 = 64
So we have,
x = 4
which leads us to
l = 32/4^2 = 32/16 = 2
These should be the dimensions of the tank to ensure
minimum use of material.
I hope its all clear, you can always get back to me.
Regards.