Question

Suppose you want to start saving for retirement. You decide to continuously invest $20000 of your income each year in a risk-free investment with a 7% yearly interest rate, compounded continuously.
If yy is the value of the investment, and tt is in years:
dydt=dydt=
Your answer should be in terms of yy.
You start investing at t=0t=0 so y(0)=0y(0)=0.
y(t)=y(t)=
What is the size of your investment after 30 years.

Answer 1

Answer:

Following are the solution to the given points:

Step-by-step explanation:

p= amount at any time

then $\frac{dp}{dt}=0.07 p$

$p=p_0 (1+e^{0.07 t)$

$P_0=\ principle \ amount \\\\ interest \ =p-p_0=p_0e^{0.07t}\\\\ p=y\frac{dy}{dt}=0.07y_0$

at time $= t\ years$

$\frac{dp}{dt}\\\\c_0=\ \ 20000\ year\\\\p \to p+dp\\\\ dp=0.07p+c_0$

y= amount at any times

$\frac{dy}{dt}=0.07y+c_0\\\\dt= \frac{dy}{c_0+0.07y}$

For point a:

$\frac{dy}{dt}=20000+0.07 y$

For point b:

$\frac{dy}{20000+0.07 y}=dt\\\\\frac{1}{0.07}|n(\frac{0.07y+20000}{20000})=t\\\\\frac{0.07y}{20000}+1=e^{0.07t}\\\\y=7.1428\times 10^4[e^{0.07t} -1]\\\\y=\frac{20000}{0.07}[e^{0.07t}-1]$

For point c:

$y(30)=827391.75$