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

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 16 years is $400,597, find the function that gives the value of the investment as a function of t. (Round your value of k to two decimal places.)

P=__________________??

What is the interest rate on this investment? (Round your answer to the nearest whole number.)

_________________%

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=16, p=400,597

400,597=180000e16k

e16k=400,597/180000

16k=ln(400,597/180000)

k=0.05

p=180000e0.05t

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

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