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

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$235,00, 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. ## Exponentially Decreasing Value of Home The median value V (in thousands of dollars) of a home in a particular market is decreasing exponentially with time t (in years). The initial value of the home is given in the question. We also know the diminished value of the home two years later. Using an exponential function to model the value of the home, we determine the value of the decay constant in the exponential function. Then we use the value of the decay constant to come up with an ordinary differential equation (ODE) along with an initial condition (IC) that makes up an initial value problem (IVP) to model the value of the home in time. ## Answer and Explanation: Let the value of the home in thousands of dollars at time t in years be given by V(t). Since the initial value of the home is$240,000 we have the exponential function model

{eq}V(t) = 240e^{-kt} \qquad (1) {/eq}

At time t = 2 years, the value of the home decreases to \$235,000 or V = 235. Using this information in (1) yields

{eq}235 = 240e^{-2k} \qquad (2) {/eq}

Dividing both sides of (2) by 240 and then taking natural logarithms of both sides leads us to

{eq}\displaystyle \ln \left( \frac {235}{240} \right) = -2k \qquad (3) {/eq}

Dividing both sides of (3) by -2 gives us k = 0.01.

Hence the ordinary differential equation (ODE) with initial condition (IC) that comprises an initial value problem (IVP) modeling the home prices is given by

{eq}\left\{ \begin{array}{l} \displaystyle \frac {dV}{dt} = -0.01V \\ \\ V(0) = 240. \end{array} \right. {/eq}