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.