Posted by **Jamie** on Tuesday, June 17, 2014 at 9:52am.

The median value of a home in a particular market is decreasing exponentially. If the value of a home

was initially $240,000, then its value two years later is $235,000.

a) Write a differential equation that models this situation. Let V represent the value of the home (in thousands of dollars) and t represent the number of years since its value was $240,000.

For this, I got dV/dt = kV but I'm not positive it's correct. I also need to solve for the particular solution in terms of V and t and I'm not sure how to do that.

- Calculus -
**Damon**, Tuesday, June 17, 2014 at 10:19am
dV/dt = k V

dV/V = k dt

ln V = k t + C

e^ln V = V definition of ln

V = e^(kt + C)

C is arbitrary so far

V = e^kt * e^C

= C e^kt since e^c could be any old C

Now that is general solution. Now put in t = 0

at t = 0 e^kt = e^0 = 1

so

240,000 = C e^0 = C

so

V = 240,000 e^kt

Now if t = 2, V = 235,000

235 = 240 e^2k

e^2k = .979

ln e^2k = 2k = -.02105

so

k = - .010526

and

V = 240,000 e^-.010526 t

- Calculus -
**Jamie**, Tuesday, June 17, 2014 at 10:26am
If the question says to write the differential equation "in thousands of dollars" would the particular solution be written as V(t)=240e^((-.010526)(t)) instead of V = 240,000 e^-.010526 t ?

- Calculus -
**Damon**, Tuesday, June 17, 2014 at 10:29am
sure, in fact I did it with 235 = 240 e^2k

- Calculus -
**Jamie**, Tuesday, June 17, 2014 at 10:34am
The last question for this problem asks to find the relative rate of change in the home's value at any time t. How would I do that?

- Calculus -
**Damon**, Tuesday, June 17, 2014 at 11:37am
You have dV/dt = k V

and you know that k = -.010526

- Calculus -
**Jamie**, Tuesday, June 17, 2014 at 11:50am
So would it be -.010526 = (dV/dt)/V ?

- Calculus -
**Damon**, Tuesday, June 17, 2014 at 12:08pm
I suppose :)

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