GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsHard
A city planner is analyzing population growth. The population, P, in thousands, can be modeled by the function P(t) = 100 * (1.02)^t, where t is the number of years since the year 2000. In what year will the population first exceed 150 thousand?
- A2024
- B2015
- C2021
- D2018
Show answer & explanationAnswer & explanation
Correct answer: C. 2021
To find when P(t) exceeds 150, set up the inequality 100 * (1.02)^t > 150. Solve for t using logarithms. (1.02)^t > 1.5. t * log(1.02) > log(1.5). t > log(1.5) / log(1.02) ≈ 20.47. Since t must be an integer year, the population exceeds 150 in the 21st year after 2000, which is 2021.
Why the other options are wrong
- A. This year is too late; the population would have already exceeded 150 thousand.
- B. This year is too early; the population has not yet reached 150 thousand.
- D. This year is too early; the population has not yet reached 150 thousand.
Solving Exponential Equations/Inequalities
To solve for a variable in an exponent, logarithms are typically used. The property log(b^x) = x * log(b) is key to bringing the exponent down.
- If b^x = y, then x = log_b(y).
- Can use common log (log₁₀) or natural log (ln).
- Property: log(A^B) = B * log(A).
- When dividing by a negative number in an inequality, flip the inequality sign.
Memory trick: Log It Down, Isolate and Solve!