Questions: Question 3 4 pts You must show ALL algebraic work to receive full credit. Amani's car was worth 15,000 at the beginning of 2010 and the value of the car decreased exponentially, decreasing by 24% each year. a. Write a function c that determines the value of the car (in dollars) in terms of the number of years y since the beginning of 2010. b. How many years after the beginning of 2010 was Amani's car worth 3,400?

Solution Steps

To solve this problem, we need to model the depreciation of Amani's car using an exponential decay function.

a. The value of the car decreases by 24% each year, which means it retains 76% of its value each year. We can express this as an exponential function where the initial value is $15,000 and the decay factor is 0.76. The function \( c(y) \) will represent the car's value after \( y \) years.

b. To find out how many years it takes for the car's value to decrease to $3,400, we need to solve the equation \( c(y) = 3,400 \) for \( y \).

The value of Amani's car after \( y \) years can be modeled by the function: \[ c(y) = 15000 \times (0.76)^y \]

To find the value of the car after 5 years, we substitute \( y = 5 \) into the function: \[ c(5) = 15000 \times (0.76)^5 \approx 3803.2881 \]

We need to solve the equation: \[ 15000 \times (0.76)^y = 3400 \] Dividing both sides by 15000 gives: \[ (0.76)^y = \frac{3400}{15000} \approx 0.2267 \] Taking the logarithm of both sides: \[ y \log(0.76) = \log(0.2267) \] Thus, we can solve for \( y \): \[ y = \frac{\log(0.2267)}{\log(0.76)} \approx 5.4084 \]

Final Answer