CALCULUS PROBLEM NEED HELP! Deals with related rates.?

Question

An item costs $200 at time t = 0 and costs $P in year t. When inflation is r% per year, the price is given by the following equation:
P= 200e ^ rt/100
a) If r is a constant, at what rate is the price rising (in dollars per year) initially? At what rate is the price rising after 2 years?

b) Now suppose that r is increasing by 0.3 per year when r = 5 and t = 2. At what rate (dollars per year) is the price increasing at that time?

Answer 1

"At what rate is the price rising?" means "what is dP/dt?"
"initially" means "at time t=0" and "after 2 years" means "for t=2"

Part (b) is more interesting. Now we are to assume that r is not constant, but instead is a function of t. Then

dP/dt = 200 [e^(rt/100)] * d/dt(rt/100) = 200[e^(rt/100)][r/100 + (dr/dt)(t/100)]

Note that this reduces to the expression you get for dP/dt in part (a) if r is a constant, because if r is a constant then dr/dt=0.

To complete this part, just evaluate dP/dt when dr/dt = 0.3, r=5, and t=2.