You buy a used car for $12,000. It depreciates at a rate of 15% per year. If this trend continues.....
a) Write an exponential decay function that models the profit in dollars over time.
b) What is the value of the car after 1 year? Write only the number with no comma.
c) What is the value of the car after 5 years? Write only the number with no comma.
if it loses 15% each year, then it retains 85%. So,
(a) y = 12000 * 0.85^x
now use that for the rest
a) Sure, here's an exponential decay function that models the profit in dollars over time for your car:
P(t) = 12000 * (0.85)^t
where P(t) represents the value of the car in dollars after t years.
b) To find the value of the car after 1 year, we can substitute t = 1 into the equation:
P(1) = 12000 * (0.85)^1
P(1) = 12000 * 0.85
P(1) = 10200
So, the value of the car after 1 year is $10,200.
c) To find the value of the car after 5 years, we can substitute t = 5 into the equation:
P(5) = 12000 * (0.85)^5
P(5) = 12000 * 0.4437053125
P(5) = 5324.46
So, the value of the car after 5 years is $5324.
a) The exponential decay function that models the profit in dollars over time can be written as:
P(t) = P₀ * e^(-r*t)
where:
- P(t) represents the value of the car after time t
- P₀ represents the initial value of the car
- r represents the rate of depreciation as a decimal (in this case, 15% or 0.15)
- t represents the time in years
b) To find the value of the car after 1 year, we can substitute the values into the function:
P(1) = $12,000 * e^(-0.15 * 1)
P(1) = $12,000 * e^(-0.15)
Using a calculator, we can evaluate e^(-0.15) ≈ 0.861
P(1) = $12,000 * 0.861
P(1) ≈ $10,332
Therefore, the value of the car after 1 year is $10,332.
c) To find the value of the car after 5 years, we can again substitute the values into the function:
P(5) = $12,000 * e^(-0.15 * 5)
P(5) = $12,000 * e^(-0.75)
Using a calculator, we can evaluate e^(-0.75) ≈ 0.472
P(5) = $12,000 * 0.472
P(5) ≈ $5,664
Therefore, the value of the car after 5 years is $5,664.
a) To write an exponential decay function that models the profit in dollars over time, we can use the formula:
P(t) = P₀ * (1 - r)^t
Where:
P(t) represents the value of the car after time t,
P₀ is the initial value of the car ($12,000 in this case),
r is the depreciation rate per year (15% or 0.15 as a decimal),
t represents the number of years.
b) To find the value of the car after 1 year, we can substitute t = 1 into the equation:
P(1) = P₀ * (1 - r)^1
P(1) = 12,000 * (1 - 0.15)^1
P(1) = 12,000 * 0.85
P(1) = 10,200
Therefore, the value of the car after 1 year is $10,200.
c) To find the value of the car after 5 years, we can substitute t = 5 into the equation:
P(5) = P₀ * (1 - r)^5
P(5) = 12,000 * (1 - 0.15)^5
P(5) = 12,000 * 0.85^5
P(5) ≈ 7,425
Therefore, the value of the car after 5 years is approximately $7,425.