Chapter 12.2, Problem 71E

Level curves of a savings plan Suppose you make monthly deposits of P dollars into an account that earns interest at a monthly rate of p%. The balance in the account after t years is $B\left(P,r,t\right)=P\left(\frac{{\left(1+r\right)}^{12t}-1}{r}\right)$, where r = p/100 (for example, if the annual interest rate is 9%, then p = $\frac{9}{12}$ = 0.75 and r = 0.0075). Let the time of investment be fixed at t = 20 years.

a.    With a target balance of $20,000, find the set of all points (P, r) that satisfy B = 20,000. This curve gives all deposits P and monthly interest rates r that result in a balance of $20,000 after 20 years.

b.    Repeat part (a) with B = $5000, $10,000, $15,000, and $25,000, and draw the resulting level curves of the balance function.