Question

A Tesla depreciates at a rate of 3% per year. If a new Tesla costs $34,000, how many years will it take until it is worth less than $30,000? Give your answer as whole years.

A) 4 years
B) 5 years
C) 6 years
D) 7 years

Answer 1

Using the formula for exponential decay, it's found that a Tesla depreciating at a 3% annual rate will be worth less than $30,000 after 5 years. The calculation involves dividing the target value by the initial cost and then using logarithms to solve for time.

Step-by-step explanation:

To calculate how many years it will take for a Tesla that depreciates at a rate of 3% per year to decrease in value from $34,000 to less than $30,000, we can use the formula for exponential decay:

V(t) = V0 × (1 - r)^t

where:

• V(t) is the value of the car after t years,
• V0 is the initial value of the car,
• r is the depreciation rate (as a decimal),
• t is the number of years.

Applying the values:

V0 = $34,000

r = 3% = 0.03

We want V(t) < $30,000, so we set up the inequality:

$30,000 > $34,000 × (1 - 0.03)^t

Solving for t, we find:

1. Divide both sides of the inequality by $34,000.
2. Take the natural logarithm of both sides to remove the exponent.
3. Solve for t.

When you follow these steps, you'll find that t exceeds 4 but does not reach 5 full years, since at the end of the fourth year the car's value will be slightly above $30,000, and at the end of the fifth year it will be below $30,000. Therefore, the answer is B) 5 years.