Calculus - Optimization Problem?
A customer wants to know the minimum cost of a water tank that will hold 750 gallons. Your company can make the tank from a rectangular piece of metal which can be rolled into a cylinder with two circular ends welded to the cylinder. The rectangular piece is made of a malleable alloy that costs $2.40 per square foot, but the circular ends can be made of a cheaper metal that costs $1.50 per square foot. Note, there are 7.5 gallons in a cubic foot. What dimensions ensure the lowest cost? What is the lowest cost?
This is pretty straightforward.
First you have to write the formula fo rthe cost of the tank as a function of, say, its radius, r:
The volume of the cylinder is 100 cubic feet.
It is also pi*r^2 times the height, h
so let's solve for h
100=pi*r^2*h
h = 100/(pi*r^2)
next, the cost:
the rectangle cost is 2.40*(h*2*pi*r),
or 2.4*100/(pi*r^2)*2*pi*r
or, simplifying, 480/r
The cost of the base and top are:
1.50*2 pieces * pi*r^2
So the total cost is
480/r+3*pi*r^2
Now you need to differentiate this and find the roots, and make sure they are minima, not maxima!
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