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 17 years is $421,136, find the function that gives the value of the investment as a function of t. (Round your value of k to two decimal places.)

Answer 1

the interest rate on this investment k = 0.05 or 5%

Explanation:

Given that;

dp/dt = kp

dp/p = k dt

now we integrate on both sides

integral dp/p = integral k dt

ln p = kt + c

p =e^kt + c

p = ce^kt

Given that p = $180,000 is invested (when t = 0)

p = ce^kt

we substitute

180000 = ce^k×0

180000 = ce^0

180000 = c × 1

c = 180,000

we substitute the value in the equation

p = 180000e^kt

now given that the amount in the account after 17 years is $421,136

i.e at t = 17 and p = 421,136

we substitute

421,136 = 180000e^17k

divide both sides by 180000

421136/180000 = 180000e^17k / 180000

e^17k = 2.3396

17k = In(2.3396)

17k = 0.8499

k = 0.04999 ≈ 0.05

therefore the interest rate on this investment k = 0.05 or 5%