Question

state the approximate doubling time formula and the conditions under which it works well. give an example.

choose the correct approximate doubling time formula for a quantity growing exponentially at a rate of p% per time period.

a. \\(t_{\text{double}} \approx 70 + p\\)
b. \\(t_{\text{double}} \approx \frac{70}{p}\\)
c. \\(t_{\text{double}} \approx \frac{p}{70}\\)
d. \\(t_{\text{double}} \approx 70 \cdot p\\)

Explanation:

Identify the doubling time formula

$$ T_{\text{double}} \approx \frac{\ln(2)}{r} \approx \frac{0.693}{r} $$

Convert rate to percentage

$$ \text{For } r = \frac{P}{100}: \quad T_{\text{double}} \approx \frac{69.3}{P} \approx \frac{70}{P} $$

Match with options

$$ T_{\text{double}} \approx \frac{70}{P} \quad \text{corresponds to Option B} $$

Answer:

• (A) \(T_{\text{double}} \approx 70 + P\)
• (B) \(T_{\text{double}} \approx \frac{70}{P}\) (Correct answer)
• (C) \(T_{\text{double}} \approx \frac{P}{70}\)
• (D) \(T_{\text{double}} \approx 70 \cdot P\)