The manufacturer's suggested retail price (MSRP) for a particular car is \( \$ 25,030 \), and it is expected to be worth \( \$ 13,994 \) in 4 years. (a) Find a linear depreciation function for this car. (b) Estimate the value of the car 7 years from now. (c) At what rate is the car depreciating?

To solve the problem, we will first establish a linear depreciation function based on the information provided. ### (a) Find a linear depreciation function for this car. Let \( V(t) \) be the value of the car at time \( t \) years, where \( t = 0 \) corresponds to the time of purchase. We know: - \( V(0) = 25,030 \) (the MSRP) - \( V(4) = 13,994 \) (the value after 4 years) We can find the slope of the linear depreciation function, which represents the rate of depreciation per year. The slope \( m \) can be calculated as follows: \[ m = \frac{V(4) - V(0)}{4 - 0} = \frac{13,994 - 25,030}{4} = \frac{-11,036}{4} = -2,759 \] Now we can write the linear depreciation function in the form: \[ V(t) = V(0) + mt \] Substituting the values we have: \[ V(t) = 25,030 - 2,759t \] ### (b) Estimate the value of the car 7 years from now. To find the value of the car after 7 years, we substitute \( t = 7 \) into the depreciation function: \[ V(7) = 25,030 - 2,759(7) \] Calculating this: \[ V(7) = 25,030 - 19,313 = 5,717 \] So, the estimated value of the car 7 years from now is **$5,717**. ### (c) At what rate is the car depreciating? The rate of depreciation is given by the slope we calculated earlier, which is: \[ \text{Rate of depreciation} = -2,759 \text{ dollars per year} \] Thus, the car is depreciating at a rate of **$2,759 per year**. ### Summary of Results: - (a) The linear depreciation function is \( V(t) = 25,030 - 2,759t \). - (b) The estimated value of the car 7 years from now is **$5,717**. - (c) The car is depreciating at a rate of **$2,759 per year**.