The figure shows a right circular cylinder with radius r and

The figure shows a right circular cylinder with radius r and height h. The volume of the cylinder is 4100 cubic feet. The bottom of the cylinder is reinforced steel, costing $85 per square foot, whereas the sides and top cost just $5 per square foot. Express the total cost of the cylinder as a function of r. Total Cost=

Solution

Volume of cylinder = pi*r^2*h

where r = radius ; h = height

So, 4100 = pi*r^2*h

h = 4100/pi*r^2

Now, bottom of the cylinder is reinforced steel, costing $85 per square foot = 85(pi*r^2)

sides and top cost just $5 per square foot = (2pi*r*h+ pi*r^2)*5

= 2pi*r*(4100/pi*r^2)*5 + 5*pi*r^2

= 41000/r +5*pi*r^2

So, Total Cost = 85*pi*r^2 + 5*pi*r^2 +41000/r

Total Cost= 90*pi*r^2 + 41000/r