First Order Differential Equation

[EV33]

Ok so we are given a word problem discussing compound interest. In the first part of the question, we are given the equation:
S(t) = (k/r)(e^rt -1)
The next thing we are asked to do is calculate the value of r are given values of k, t, and
S(t). The given values are k = 2000, t = 40, S(t) = 10^6.
Plugging the values into this equation, you get:
10^6 = (2000/r)(e^40r - 1)

So far I have divided both sides by 2000 so that the equation is now:
500 = (1/r)(e^40r -1)

However now I can not figure out how to solve the equation for r with the two locations of the r's in the problem. Please help.

Thanks.

 

[Dick]

You can't solve that using algebra and simple functions. You'll have to solve it numerically. Or graph the function and figure out where it crosses 500. I'd concentrate near small numbers like r=0.01.

 

[EV33]

What do you mean by solving it numerically?

 

[Dick]

I mean if f(r)=(2000/r)(e^40r - 1) if I put r=0.01, I get 98364.93. If I put r=0.1 I get 1071963.00. So the r such that f(r)=10^6 must be somewhere in between. Probably a lot closer to 0.1. Refine your guess and keep closing in on the answer. That's a 'numerical technique'. The problem doesn't isn't solvable in any simple way. You'll have to settle for an approximation.