To determine how long it will take for the car to be worth $8,200 using the given exponential decay model V = 20,000(0.8)^x:
1. Set up the equation with the given value of the car: 8,200 = 20,000(0.8)^x.
2. Divide both sides by 20,000 to isolate the exponential term: 8,200 / 20,000 = (0.8)^x.
3. Simplify the left side: 0.41 = (0.8)^x.
4. To find the value of x (the number of years), take the logarithm of both sides to solve for x: log(0.41) = log((0.8)^x).
5. Apply the logarithmic property log(a^b) = b*log(a) to get x out of the exponent: x = log(0.41) / log(0.8).
6. Calculate the value of x using a calculator to find approximately how long it will take for the car to be worth $8,200.
By following these steps and performing the calculations, you can determine the approximate number of years it will take for the car's value to reach $8,200.
