Question

the formula for depreciation is \\(v = c(1 - r)^t\\), where \\(v\\) is the value of the car after \\(t\\) years, \\(c\\) is the original cost, and \\(r\\) the rate of depreciation (as a decimal). steven found that the value of his new car is currently \\(\\$31,200\\), but it is expected to depreciate \\(6.9\\%\\) per year. what will be the value of his car in 5 years?

a. \\(\\$29,432\\)

b. \\(\\$24,040\\)

c. \\(\\$21,822\\)

d. \\(\\$31,200\\)

Explanation:

Identify the given variables

We extract the values from the problem statement:

• Original cost \(C = 31,200\)
• Depreciation rate \(r = 6.9\% = 0.069\)
• Time in years \(t = 5\)
• Depreciation formula:

$$V = C(1 - r)^t$$

Calculate the decay factor

Using the Algebraic Simplification knowledge point:

$$1 - r = 1 - 0.069 = 0.931$$

Evaluate the exponential term

Using the Exponent Rules knowledge point

$$ (0.931)^5 \approx 0.700064 $$

Calculate the final value

Using the Algebraic Evaluation knowledge point

$$ V = 31,200 \times 0.700064 \approx 21,841.99 $$

Comparing this result to the given options, the closest value is \(\$21,822\) (Option C), which corresponds to slight rounding variations in intermediate steps.

Verify the closest option

Let's double check the calculation:

• If \(1 - r = 0.931\)
• \(31200 \times (0.931)^5 \approx 21841.99\)
• Let's check if there is a typo in the options or a slight difference in rounding. Option C is \(\$21,822\), which is extremely close and the only reasonable choice near \(\$21.8\text{k}\).

Answer:

• (A) \$29,432
• (B) \$24,040
• (C) \$21,822 (Correct answer)
• (D) \$31,200