GED Mathematical Reasoning TestAlgebraic Problem Solving with Expressions and EquationsHard
A biologist is tracking the population of a certain type of algae in a pond. The population, P(t), in thousands, after 't' days, is modeled by the function P(t) = 50e^(0.08t). Approximately how many days will it take for the algae population to reach 150 thousand?
- A8 days
- B18 days
- C22 days
- D14 days
Show answer & explanationAnswer & explanation
Correct answer: D. 14 days
We need to solve 150 = 50e^(0.08t) for t. First, divide both sides by 50: 3 = e^(0.08t). To isolate 't', take the natural logarithm (ln) of both sides: ln(3) = ln(e^(0.08t)). Using the logarithm property ln(a^b) = b*ln(a), we get ln(3) = 0.08t * ln(e). Since ln(e) = 1, the equation becomes ln(3) = 0.08t. Calculate ln(3) ≈ 1.0986. So, 1.0986 = 0.08t. Divide by 0.08: t = 1.0986 / 0.08 ≈ 13.7325. Approximately 14 days.
Why the other options are wrong
- A. This is too low; it might result from incorrect logarithmic calculations or algebraic errors.
- B. This is too high; it could be a result of calculation errors or misinterpreting the natural logarithm.
- C. This is significantly too high and likely results from substantial calculation errors or incorrect use of the natural logarithm.
Solving Exponential Equations (ln)
To solve exponential equations where the base is 'e' or a general base, use logarithms. The natural logarithm (ln) is particularly useful for base 'e' because ln(e^x) = x.
- Isolate the exponential term (e^(kt)).
- Take the natural logarithm (ln) of both sides.
- Use the property ln(a^b) = b*ln(a) to bring the exponent down.
- Solve for the variable.
Memory trick: Isolate 'e', then 'ln' to free the 't'.