Question

Quentin has opened a retirement fund in which he will deposit 15% of his salary each year. If his current salary is $70,000 per year and he expects that it will increase 5% each year, what will be the present worth of the fund after 30 years if it earns a 7% interest rate per year?(10 pts)

Answer 1

The rate of change future value is

dV dV15 70000e0.05 0.07V 100 一0.071-10500e0.05t dt

Solve this differential equation,

$\frac{\mathrm{d} \left (V e^{-0.07t } \right )}{\mathrm{d} t}=10500e^{0.05t}e^{-0.07t}\\ \int \frac{\mathrm{d} \left (V e^{-0.07t } \right )}{\mathrm{d} t}dt=\int 10500e^{0.05t}e^{-0.07t}dt\\ V e^{-0.07t } =10500\int e^{-0.02t}dt\\ V e^{-0.07t } =-\frac{10500}{0.02}e^{-0.02t}+C\\ V =-525000e^{0.05t}+Ce^{0.07t}$

At .

$0 =-525000e^{0}+Ce^{0}\Rightarrow C=525000$

So at $t=30$ , the worth of the fund is

$V\left ( 30 \right ) =-525000e^{0.05\times 30}+525000e^{0.07\times 30}\\ {\color{Blue} V\left ( 30 \right ) =\$1,934,352.44}$