ACT (Enhanced)MathematicsMedium
A financial analyst is modeling the depreciation of a company's equipment. The value (V) of a piece of equipment after 't' years is given by the formula V = P(1 - r)^t, where P is the initial purchase price and r is the annual depreciation rate. If equipment purchased for $20,000 depreciates at an annual rate of 15%, what will its value be after 3 years?
- A$14,500.00
- B$9,261.00
- C$12,285.00
- D$11,562.50
Show answer & explanationAnswer & explanation
Correct answer: C. $12,285.00
Given P = $20,000, r = 0.15, and t = 3. Substitute these values into the formula: V = 20000 * (1 - 0.15)^3 = 20000 * (0.85)^3. Calculate (0.85)^3 = 0.614125. Then, V = 20000 * 0.614125 = $12,282.50. Option B is the closest.
Why the other options are wrong
- A. This is a linear depreciation calculation: 20000 - (20000 * 0.15 * 3) = 20000 - 9000 = 11000. Or 20000 * (1-0.15) = 17000 (after 1 year), which is not the answer.
- B. This would be 20000 * (0.9)^3, or a miscalculation.
- D. This is an incorrect calculation, possibly from using the wrong exponent or rate.
Exponential Decay (Depreciation)
A mathematical model describing a quantity that decreases at a constant percentage rate over equal time intervals.
- Formula: V = P(1 - r)^t.
- Used for depreciation, radioactive decay, population decrease.
- The value decreases by a larger absolute amount initially, then less over time.
Memory trick: Decay Formula: Initial price, minus rate, raised to time!