The manufacturer's suggested retail price (MSRP) for a particular car is \( \$ 24,605 \), and it is expected to be worth \( \$ 10,970 \) in 5 years. (a) Find a linear depreciation function for this car. (b) Estimate the value of the car 6 years from now. (c) At what rate is the car depreciating?

Given: - The manufacturer's suggested retail price (MSRP) for a particular car is $24,605. - The car is expected to be worth $10,970 in 5 years. (a) To find a linear depreciation function for this car, we can use the formula for linear depreciation: \[ V(t) = V_0 - rt \] where: - \( V(t) \) is the value of the car at time \( t \), - \( V_0 \) is the initial value of the car (MSRP), - \( r \) is the rate of depreciation, - \( t \) is the time in years. We are given that the car is expected to be worth $10,970 in 5 years. This means that at \( t = 5 \), the value of the car is $10,970. We can use this information to find the rate of depreciation \( r \). Let's calculate the rate of depreciation \( r \) first. Solve the equation by following steps: - step0: Solve for \(r\): \(24605-5r=10970\) - step1: Move the constant to the right side: \(-5r=10970-24605\) - step2: Subtract the numbers: \(-5r=-13635\) - step3: Change the signs: \(5r=13635\) - step4: Divide both sides: \(\frac{5r}{5}=\frac{13635}{5}\) - step5: Divide the numbers: \(r=2727\) The rate of depreciation \( r \) is $2727 per year. Now, we can find the linear depreciation function for the car: \[ V(t) = 24,605 - 2727t \] (b) To estimate the value of the car 6 years from now, we can substitute \( t = 6 \) into the linear depreciation function: \[ V(6) = 24,605 - 2727(6) \] Let's calculate the value of the car 6 years from now. Calculate the value by following steps: - step0: Calculate: \(24605-2727\times 6\) - step1: Multiply the numbers: \(24605-16362\) - step2: Subtract the numbers: \(8243\) The estimated value of the car 6 years from now is $8243. (c) To find the rate at which the car is depreciating, we can use the rate of depreciation \( r \) we calculated earlier: \[ r = \$2727 \text{ per year} \] Therefore, the car is depreciating at a rate of $2727 per year.