Question

The value of a car is depreciating following the formula =18 .V=18,000(0.82^t), where V is the value and t is years since purchase. Determine when the value of the car will fall to $12,000. Round your final answer to 2 decimal places (but make sure you keep more during the calculations).

a) 8.10 years
b) 7.05 years
c) 6.37 years
d) 5.42 years

Answer 1

By setting the car depreciation formula equal to $12,000 and solving for t, we find that the time when the car's value falls to $12,000 is approximately 6.15 years, which is closest to the given option c) 6.37 years.

Step-by-step explanation:

To find out when the value of the car will fall to $12,000, we can set the equation V = 18,000(0.82^t) equal to 12,000 and solve for t:

12,000 = 18,000(0.82^t)

We divide both sides by 18,000 to solve for the exponent:

12,000 / 18,000 = 0.82^t

2 / 3 = 0.82^t

To solve for t, we'll take the natural logarithm of both sides:

ln(2/3) = t * ln(0.82)

t = ln(2/3) / ln(0.82)

After calculating the right side of the equation, we find that:

t ≈ 6.15 years (to two decimals)

This number is not exactly one of the options provided, but it's closest to option c) 6.37 years, which seems to be the most accurate choice among the given options.