It's a differential equation of sorts, but beyond that....im not sure
what else to do.

Because of a slump in the economy, a company finds that its annual
revenue has  dropped from 742,000  in 1998 to 632,000 in 2000. if the
revenue is following  an exponential pattern of decline, what is the
expected revenue for 2002? (t=0 in 1998)

Hello!
If revenue follows an exponential pattern of decline, we should be
able to characterize it with a function of the form:

Revenue(t) = A * e^(c * t)

where:
t is time
c is a decay parameter
A is the level of sales at t = 0

So, first of all, why does A represent the revenue at t = 0? If we
plug t = 0 into the equation, we get:

Revenue(0) = A * e^(c * 0)
Revenue(0) = A * e^0
Revenue(0) = A * 1
Revenue(0) = A

Therefore, we've found that in the Revenue(t) equation, A = 742,000
(the revenue of the company when t = 0)

We know that 2 years later (at t = 2), revenue has become 632,000. So
now we can use this information to find c, the missing parameter:

Revenue(2) = 742000 * e^(c * 2)
632000   = 742000 * e^(c * 2)
632/742  = e^(c * 2)
ln(632/742) = c * 2
c = ln(632/742) / 2
c = -0.0802299...

Therefore, we've found that the revenue function is:

Revenue(t) = 742000 * e^(-0.0802299 * t)

Now, what's the expected revenue for year 2002? In this case, t = 4.
Therefore, we simply plug t = 4 into our equation, getting:

Revenue(4) = 742000 * e^(-0.0802299 * 4)
Revenue(4) = 538307.33

Expected revenue for year 2002 is thus 538,307.33


I hope this helps! If you have any doubt regarding my answer, please
don't hesitate to request clarification before rating it. Otherwise, I
await your rating and final comments.

Best wishes!
elmarto

Homework???

It's actually part of a study guide for a final exam

A much simpler way to think of it would be to simply calculate the
compound annual growth rate. [ (Current value / Previous value) ^ ( 1
/ n-years) -1 ]

Therefore, from 1998 to 2000 the annualized rate is -7.7%.  Using this
figure, the revenue estimate for 2002 is $538.3K.