A hospital trust decides to install a new rooftop solar power system which will save money on its energy bills. It estimates that t years after installation the savings flow (in $1000s per year) will steadily increase according to S(t) = 0.3t + 45.6
The flow of money needed to repair and maintain the system in subsequent years is modelled by C(t) = 0.5t² + 2t
(a) Draw a sketch graph of the savings and cost functions on the same diagram.
(b) Solve the equation S(t) = C(t) to determine the period during which the savings exceed the maintenance costs, and shade the area which determines the net savings during this time.
(c) Use integration to find the exact value of net savings in part (b).