Respuesta :
Answer:
The maintance cost and the value will be the same in 9 years.
Step-by-step explanation:
Exponential function:
An exponential function has the following format:
[tex]y(t) = y(0)(1 + r)^t[/tex]
In which r is the rate of change.
The value starts at $20,000 and decreases by 15% each year.
This means that [tex]V(0) = 20000, r = -0.15[/tex]
So
[tex]V(t) = V(0)(1 + r)^t[/tex]
[tex]V(t) = 20000(1 - 0.15)^t[/tex]
[tex]V(t) = 20000(0.85)^t[/tex]
The maintance cost is $500 the first year and increases by 28% per year.
This means that [tex]M(0) = 500, r = 0.28[/tex]. So
[tex]M(t) = M(0)(1 + r)^t[/tex]
[tex]M(t) = 500(1 + 0.28)^t[/tex]
[tex]M(t) = 500(1.28)^t[/tex]
When will the maintenance cost and the value be the same.
This is t for which:
[tex]V(t) = M(t)[/tex]
So
[tex]20000(0.85)^t = 500(1.28)^t[/tex]
[tex]\frac{(1.28)^t}{(0.85)^t} = \frac{20000}{500}[/tex]
[tex](\frac{1.28}{0.85})^t = 40[/tex]
[tex]\log{(\frac{1.28}{0.85})^t} = \log{40}[/tex]
[tex]t\log{(\frac{1.28}{0.85})} = \log{40}[/tex]
[tex]t = \frac{\log{40}}{\log{(\frac{1.28}{0.85})}}[/tex]
[tex]t = 9[/tex]
The maintance cost and the value will be the same in 9 years.
