The value of a new car purchased for $28,000 decreases 8% per year. Write an exponential decay model for the value of the car. Use the model to estimate the value after 5 years.
To write an exponential decay model for the value of the car, we can use the formula:
V = P(1 - r)^t
Where:
- V is the value of the car after t years
- P is the initial purchase price of the car ($28,000 in this case)
- r is the rate of decrease as a decimal (8% is 0.08)
- t is the number of years
Substituting the values in, we get:
V = 28000(1 - 0.08)^5
Now, let's calculate the value of the car after 5 years.
To write an exponential decay model for the value of the car, we need to determine the initial value and the decay rate.
Given that the new car is purchased for $28,000, the initial value is $28,000.
The car decreases in value by 8% per year. In other words, the decay rate is 8% or 0.08 in decimal form.
Now we can write the exponential decay model:
V(t) = V₀ * (1 - r)^t
V(t) represents the value of the car after t years,
V₀ is the initial value of the car,
r is the decay rate or percentage decrease (in decimal form), and
t is the number of years.
Plugging in the values we have:
V(t) = 28,000 * (1 - 0.08)^t
Now, to estimate the value of the car after 5 years, we substitute t = 5 in the equation:
V(5) = 28,000 * (1 - 0.08)^5
Calculating this expression will give us the estimated value of the car after 5 years.
Let P(n) is the price after n years, then
P(n)=28000*0.92^n
P(5)=28000*0.92^5=18454