Suppose that t years from now, one investment plan will be generating profit at

Question

Suppose that t years from now, one investment plan will be generating profit at the rate of P1(t)=50e^(0.1t) thousand dollars per year, while a second investment will be generating P2(t)=160e^(0.04t) thousand dollars per year.

a. For how many years does the rate of profitability of the second investment exceed that of the first?

Answer:______

b. Compute the net excess profit, in thousands of dollars, assuming that you invest in the second plan for the time period determined in part a.

Answer:______

Explanation / Answer

given P1(t)=50e0.1t

P2(t)=160e0.04t

a) P1(t)= P2(t)

50e0.1t =160e0.04t

e0.1t /e0.04t=160/50

e0.1t-0.04t=16/5

e0.06t =16/5

0.06t =ln(16/5)

t =(1/0.06)ln(16/5)

t =19.4 years

For 19.4 years does the rate of profitability of the second investment exceed that of the first

b)net excess profit =[0 to 19.4]160e0.04t - 50e0.1t dt

net excess profit =[0 to 19.4](160/0.04)e0.04t -(50/0.1)e0.1t

net excess profit =[0 to 19.4](4000)e0.04t -(500)e0.1t

net excess profit =(4000)e0.04*19.4 -(500)e0.1*19.4 -(4000)e0 +(500)e0

net excess profit =1711.7 thousand dollars

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