Hi.. Could you help me with solving this problem? 40. Paying Down a Mortgage You

Question

Hi..

Could you help me with solving this problem?

40. Paying Down a Mortgage You took a $200,000 home mortgage at an annual interest rate of 3%. Suppose that the loan is amortized over a period of 30 years, and let P(t) denote the amount of money (in thousands of dollars) that you owe on the loan after t years. A reasonable estimate of the rate of change of P is given by P (t) = ?4.1107e0.03t . (a) Approximate the net change in P after 20 years. (b) What is the amount of money owed on the loan after 20 years? (c) Verify that the loan is paid off in 30 years by computing the net change in P after 30 years.

Explanation / Answer

solution:

P(t)=integrate(-4110.7*e^(0.03t) w.r.t t)=-4110.7/0.03*e^(0.03t)+c

At t=0: P(t)=200000

=>c=337023.33

1.

at t=20: P'(t)=-4110.7*e^(0.03*20) =-7490.18

2.

Amount of money owed after 20 years: P(t)=-4110.7/0.03*e^(0.03t)+337023.33=-4110.7/0.03*e^(0.03*20)+337023.33=87350.54

3.

at t=30: P'(t)=-4110.7*e^(0.03*30) =-10110.7

As change in loan outstanding is turning negative, the loan has been repaid

ALso,

P(t)=-4110.7/0.03*e^(0.03t)+337023.33=-4110.7/0.03*e^(0.03*30)+337023.33=0.31032

We see that the loan outstnaidn is close to zero.

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