Question

Suppose that the dollar value v(t) of a certain house that is t years old is given by the following exponential function.v(t) = 559,900(0.81)Find the initial value of the house.Does the function represent growth or decay?By what percent does the value of the house change each year?

Answer 1

Given:

`v(t)=559,900(0.81)^t`

To determine the initial value of the house, we note first that the general form for the exponential decay formula is:

`f(x)=a(1-r)^x`

where:

a=initial amount

1-r=decay factor

x= time period

So based on the given exponential function of the house, the initial value of the house is $ 559,900.

Next, we can say that the given function represent decay because the decay factor is 0.81 which is greater than zero but less than 1.

Now, to determine what percent does the value of the house change each year, we do the steps below:

`\begin{gathered} 1-r=0.81 \\ \text{Simplify} \\ r=1-0.81 \\ r=0.19 \\ r=19\% \end{gathered}`

Hence, the value of the house change each year is 19%.