The function \(\mathrm{f(t) = 40{,}000(2)^{(t/790)}}\) gives the number of bacteria in a population t minutes after an initial observation. How...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(t) = 40,000(2)^{(t/790)}}\) represents bacteria population after t minutes
• Find: Time for population to double

2. INFER what doubling means mathematically

• If population doubles, then \(\mathrm{f(t) = 2 \times f(0)}\)
• Initial population: \(\mathrm{f(0) = 40,000(2)^0 = 40,000}\)
• Doubled population: \(\mathrm{2 \times 40,000 = 80,000}\)

3. TRANSLATE doubling into an equation

• We need: \(\mathrm{40,000(2)^{(t/790)} = 80,000}\)
• Dividing both sides by 40,000: \(\mathrm{(2)^{(t/790)} = 2}\)

4. INFER the key insight about exponential doubling

• For \(\mathrm{2^{(t/790)}}\) to equal 2, we need: \(\mathrm{2^{(t/790)} = 2^1}\)
• Since bases are equal, exponents must be equal: \(\mathrm{t/790 = 1}\)

5. SIMPLIFY to find the answer

• From \(\mathrm{t/790 = 1}\), we get \(\mathrm{t = 790}\)

Answer: B. 790

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students focus on the initial value (40,000) rather than understanding that doubling is controlled by the exponential factor \(\mathrm{2^{(t/790)}}\). They might think the answer involves 40,000 in some way or that they need to double something related to the initial population.

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about exponential growth: Students may not recognize that in exponential functions, the time for doubling depends only on the base and growth rate structure, not the initial amount. They might incorrectly think the doubling time should be related to the coefficient 40,000.

This may lead them to select Choice D (40,000) or causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand that exponential doubling time depends solely on when the exponential growth factor itself doubles (\(\mathrm{2^{(t/790)} = 2}\)), which is a fundamental property of exponential functions that's independent of the initial population size.