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JustCurious' question at Yahoo! Answers regarding a mathematical model
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Hello JustCurious,
If we let $L$ be the balance in dollars left on the loan at time $t$ in years, and $r$ be the annual interest rate, then from the given information, we may model the loan balance with the following IVP:
\(\displaystyle \frac{dL}{dt}=rL-k\) where \(\displaystyle L(0)=L_0\)
Separating variables, and switching dummy variables of integration to allow using the boundaries as limits, we obtain:
\(\displaystyle \int_{L_0}^L\frac{1}{ru-k}\,du=\int_0^t\,dv\)
Applying the anti-derivative form of the FTOC, we get:
\(\displaystyle \ln\left|\frac{rL-k}{rL_0-k} \right|=rt\)
Converting from logarithmic to exponential form, we have:
\(\displaystyle \frac{rL-k}{rL_0-k}=e^{rt}\)
Solving for $k$, there results:
\(\displaystyle k=\frac{r\left(L_0e^{rt}-L \right)}{e^{rt}-1}\)
Plugging in the desired data:
\(\displaystyle r=\frac{1}{10},\,L_0=8000,\,t=3,\,L=0\)
we obtain:
\(\displaystyle k=\frac{800}{1-e^{-\frac{3}{10}}}\approx3086.636730808066\)
Hence, we find that the annual rate of repayment is about $3,086.64.
If we let $L$ be the balance in dollars left on the loan at time $t$ in years, and $r$ be the annual interest rate, then from the given information, we may model the loan balance with the following IVP:
\(\displaystyle \frac{dL}{dt}=rL-k\) where \(\displaystyle L(0)=L_0\)
Separating variables, and switching dummy variables of integration to allow using the boundaries as limits, we obtain:
\(\displaystyle \int_{L_0}^L\frac{1}{ru-k}\,du=\int_0^t\,dv\)
Applying the anti-derivative form of the FTOC, we get:
\(\displaystyle \ln\left|\frac{rL-k}{rL_0-k} \right|=rt\)
Converting from logarithmic to exponential form, we have:
\(\displaystyle \frac{rL-k}{rL_0-k}=e^{rt}\)
Solving for $k$, there results:
\(\displaystyle k=\frac{r\left(L_0e^{rt}-L \right)}{e^{rt}-1}\)
Plugging in the desired data:
\(\displaystyle r=\frac{1}{10},\,L_0=8000,\,t=3,\,L=0\)
we obtain:
\(\displaystyle k=\frac{800}{1-e^{-\frac{3}{10}}}\approx3086.636730808066\)
Hence, we find that the annual rate of repayment is about $3,086.64.