Question

the value of a house in 2015 was 400000. its value has been doubling each decade.1. if v is the value of the home, in dollars, write an equation for v in terms of d, the number of decades since 2015. 2. What is v when d = -1? What does this value mean.3. What is v when d = -3? What does this value mean.

Answer 1

We know that the value of the house doubles each decade.

Then, the value can be written as:

$V(d)=2\cdot V(d-1)$

where d is the actual decade, and (d-1) is the previous decade.

If d=0 is the decade of 2015, we have:

$\begin{gathered} V(0)=400,000 \\ V(1)=2\cdot V(0)=2\cdot400,000 \\ V(2)=2\cdot V(1)=2\cdot(2\cdot V(0))=2^2\cdot V(0)=2^2\cdot400,000 \\ \Rightarrow V(d)=2^d\cdot400,000 \end{gathered}$

We have the explicit formula of the value as:

$V(d)=400,000\cdot2^d$

Now we can use this formula to calculate the value of the house for any decade.

When d=-1, the value is:

$V(-1)=400,000\cdot2^{-1}=\frac{400,000}{2}=200,000$

When d=-2, the value is:

$V(-2)=400,000\cdot2^{-2}=\frac{400,000}{2^2}=\frac{400,000}{4}=100,000$

Answer:

1. V(d) = 400,000 * 2^d

2. V(-1) = 200,000

3. V(-3) = 100,000