Suppose that when it is t years old, a particular industrial machine generates revenue at the rate R'(t) = 6,025 – 8t² dollars per year, while operating and maintenance costs accumulate at the rate C'(t) = 4,681 + 13t² dollars per year.

a. The useful life of a machine is the number of years T before the profit it generates starts to decline. What is the useful life of this machine?

b. Compute the net profit generated by the machine over its useful lifetime.

c. Sketch the revenue rate curve y = R'(t) and the cost rate curve y = C'(t), and shade the region whose area represents the net profit computed in part (b).