Bob bought some land costing $16,240. Today, that same land is valued at $46,517. How long has Bob owned this land if the price of land has been increasing at 6 percent per year?

Question

BlackTom AI · Accepted Answer

Use the compound-growth formula FV = PV(1 + r)^t, because land value growing at a fixed percent per year follows exponential growth.

1. Relevant concepts
- Exponential (compound) growth: future value FV = present value PV × (1 + r)^t, where r is the annual growth rate and t is time in years.
- Solve for time: t = ln(FV / PV) / ln(1 + r).

2. Step-by-step reasoning
- Given PV = $16,240, FV = $46,517, and r = 6% = 0.06.
- Compute the ratio FV / PV:
  FV / PV = 46,517 / 16,240 ≈ 2.8645.
- Take natural logarithms and divide by ln(1.06) to isolate t:
  t = ln(2.8645) / ln(1.06).

3. Intermediate calculations
- ln(2.8645) ≈ 1.0522 (since e^1.0522 ≈ 2.8645).
- ln(1.06) ≈ 0.0582689.
- t ≈ 1.0522 / 0.0582689 ≈ 18.058 ≈ 18.06 years.

4. Why this answer is correct
- Plugging t back verifies the result: (1.06)^18.06 ≈ 2.8645, and 16,240 × 2.8645 ≈ 46,517, matching the given future value. The algebraic derivation and numerical check confirm the time of ownership is about 18.06 years.

5. Why the other options are incorrect (brief)
- 16.22 years: (1.06)^16.22 ≈ 2.573 → FV ≈ $41,790, too low.
- 17.19 years: (1.06)^17.19 ≈ 2.722 → FV ≈ $44,260, too low.
- 19.09 years: (1.06)^19.09 ≈ 3.042 → FV ≈ $49,460, too high.
- 20.24 years: (1.06)^20.24 ≈ 3.252 → FV ≈ $52,841, too high.
Each of these produces a future value different from $46,517, so they are not correct.

6. Conclusion
The calculated time of ownership is 18.06 years, which matches the provided correct answer.

Bob bought some land costing $16,240. Today, that same land is valued at $46,517. How long has Bob owned this land if the price of land has been increasing at 6 percent per year?Single choice

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