A scientist is tracking the population of a certain type of bacteria in a petri dish. The population, P, after 't' hours, is given by the equation P = 200 * (1.5)^t. If the population reaches 675 bacteria, how many hours have passed?
- A2 hours
- B4 hours
- C3 hours
- D5 hours
Show answer & explanationAnswer & explanation
Correct answer: C. 3 hours
Substitute P = 675 into the equation: 675 = 200 * (1.5)^t. First, divide both sides by 200: 675 / 200 = (1.5)^t, which simplifies to 3.375 = (1.5)^t. To solve for t, we can use logarithms. Take the logarithm of both sides (natural log or base-10 log): ln(3.375) = ln((1.5)^t). Using the logarithm property ln(a^b) = b * ln(a), we get ln(3.375) = t * ln(1.5). Now, solve for t: t = ln(3.375) / ln(1.5). Using a calculator, ln(3.375) ≈ 1.216 and ln(1.5) ≈ 0.405. So, t ≈ 1.216 / 0.405 ≈ 3. Therefore, approximately 3 hours have passed.
Why the other options are wrong
- A. This results from an incorrect calculation or misapplication of logarithms, possibly finding t for a smaller growth factor.
- B. This results from an incorrect calculation or conceptual error, possibly assuming linear growth.
- D. This results from an incorrect calculation, possibly multiplying by 1.5 instead of raising to the power of t.
Solving Exponential Equations (Logarithms)
Finding the value of a variable in an exponent by applying logarithms to both sides of the equation, utilizing logarithm properties.
- Isolate the exponential term first.
- Take log (natural or common) of both sides.
- Use log property: log(a^b) = b*log(a).
Memory trick: Isolate, Log, Divide: Get the base alone, then use logs to find the power!