compounded interest

[mathstresser]

Clarissa wants to buy a new car. Her loan officer tells her that her annual rate is 8%, compounded continuously, over a four-year term. Clarissa informs her loan officer that she can make equal monthly payments of $225. How much can Clarissa afford to borrow?

P’=rP-w =>

P(t)= Ae^(rt) +w/r
or
P(t)= (P+w/r)e^(rt) -w/r

r=.08
w=225
t=4

I get:

P(t)= (P +225/.08)e^(.08*4)-225/.08
P(t)= (P+2812.5)e^(.32)-2312.5

Is that right so far?

What else do I need to do?

 

[mark07]

P’=rP-w =>

P(t)= Ae^(rt) +w/r -> Correct

or
P(t)= (P+w/r)e^(rt) -w/r -> Wrong

Assuming P(0)=P (loan amount), you should get

\(\displaystyle \L P(t)= \left( P-\frac{w}{r}\right) e^{rt} + \frac{w}{r}\)

Also, time is measured in years, so w is not 225, it is 12*225.

Question is to find P. You can find it by setting the loan amount at time 4 to be 0.

i.e. solve P(4)=0 for P,

\(\displaystyle \L 0 = P(4)= \left( P-\frac{(12)(225)}{0.08}\right) e^{(4)(0.08)} + \frac{(12)(225)}{0.08}\)

I get P = 9242.47.