When the interest on an investment is compounded continuously, the investment gr

Question

When the interest on an investment is compounded continuously, the investment grows at a rate that is proportional to the amount in the account, so that if the amount present is P, then

dp/dt = kP

where P is in dollars, t is in years, and k is a constant. If $180,000 is invested (when t = 0) and the amount in the account after 10 years is $296,820, find the function that gives the value of the investment as a function of t.

P= ___________________???

What is the interest rate on this investment? (Round to the nearest integer.)

___________%

Explanation / Answer

dp/dt =kp

dp/p=k dt

integrate on both sides

integral dp/p=integral k dt

ln p =kt +c

p=ekt +c

p=cekt

at t=0

180000=cek*0

c=180000

p=180000ekt

at t=10, p=296,820

296,820=180000e10k

e10k=296,820/180000

10k=ln(296,820/180000)

k=0.05

p=180000e0.05t

interest rate on this investment=k=0.05=5.00%

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