Bob bought some land costing $14,990. Today that same land is valued at $55,000. How long has Bob owned this land if the price of land has been increasing at 6 percent per year?

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BlackTom AI · Accepted Answer

We start by translating the situation into a compound growth problem. If land value grows at 6% per year, then after t years the value is given by 14990 × (1.06)^t, and we are told this equals 55000.

Option A: 19.82 years. If we plug t = 19.82 into 14990 × (1.06)^t, the resulting value would be significantly less than 55000 because (1.06)^t grows with t, but 19.82 is too small to reach 55000 starting from 14990 at 6% annual growth. This indicates this t is not enough time for the price to reach 55000.

Option B: 20.98 years. This is closer but still yields a value under 55000 when computed with 14990 × (1.06)^{20.98}. While the result increases with t, 20.98 years does not produce exactly 55000; it remains short of the target, showing this is not the precise solution.

Option C: 21.03 years. Similarly, at 21.03 years the computed value using the formula would be closer than 20.98 but still would not match 55000 exactly. The discrepancy shows this is not the correct time span for which the value hits 55000 at 6% growth.

Option D: 22.31 years. This is the time that makes 14990 × (1.06)^t equal to 55000 when t ≈ ln(55000/14990) / ln(1.06). Carrying out the calculation: ratio 55000/14990 ≈ 3.668, ln(ratio) ≈ 1.299, ln(1.06) ≈ 0.05827, so t ≈ 1.299 / 0.05827 ≈ 22.31 years. This aligns with the required exponential growth to reach 55000 from 14990 at 6% annually.

In summary, the correct approach is to set up the exponential growth equation and solve for t using logarithms. The other options correspond to times that do not satisfy the equation 14990 × (1.06)^t = 55000, hence they underestimate the required duration to reach 55000 at 6% annual growth.

MSB-250-300-002 Proctored Midcourse Exam 1

Bob bought some land costing $14,990. Today that same land is valued at $55,000. How long has Bob owned this land if the price of land has been increasing at 6 percent per year?

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