Max Mortgage Borrowing & Interest Paid for $800/mo - ODE Modeling Problem

[amcavoy]

"A home buyer can afford to spend no more than $800/month on mortgage payments. Suppose that the interest rate is 9% and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. 1) determine the maximum amount that this buyer can afford to borrow. 2) determine the total interest paid during the term of the mortgage."

The first thing I did was to find out the total amount paid (including interest) after 20 years. I came up with 240months*$800/month=$192,000. Using this, I know know that the answer to #2 will be 192,000-ans(1). However, I seem to be making a mistake in setting up the ODE for question #1. Let S be the amount owed:

$$\frac{dS}{dt}=.09S-800$$

The reason I set it up like so is because to me, it seemed like for each payment made, 9% of the amt. owed at that point would go towards interest and the rest would come off the current amt. I know how to solve these fine, I just need some help setting it up. Am I on the right track with my model above (I know it's not correct)?

I appreciate it.

 

[mathmike]

p(t) = pe^(rt)

where p = initial amount

t = time

r = interest rate

 

[amcavoy]

p(t) = pe^(rt)

where p = initial amount

t = time

r = interest rate

192,000=P₀e^.09*20

Solving I come up with P₀=$31,737.39, which is incorrect.

Where have I gone wrong? Thanks for the help!

 

[amcavoy]

Nevermind I should have replaced 800 with 9600 in my original ODE.