Find an expression that represents the (total) cost to build the curved roof as a function of the radius r of the semicircular end.

Question

A wind tunnel is formed of a half cylinder with a rectangular base and semicircular ends. The volume must be 10000 ft3 and the curved roof costs $7 per ft2 to build. Find an expression that represents the (total) cost to build the curved roof as a function of the radius r of the semicircular end.

Answer 1

The volume enclosed by the roof = (1/2)πr²(L) = 10,000, where r is the radius of the semicircular end and L is the length of the roof.  

 

So, L = 20,000/(πr²)

 

Surface area of roof = (1/2)(2πr)(L) = (1/2)(2πr)(20,000)/(πr²)

                                                                   = 20,000/r

Cost of roof = 7(surface area)

                      = 140,000/r