Question

The growth of population is proportional to the number present. If the population of a colony doubles in 50 years, then the population will become triple in

• A. \( 5 \cdot \frac{\log 2}{\log 3} \) years
• B. \( 50 \cdot \frac{\log 3}{\log 2} \) years
• C. \( 5 \cdot \frac{\log 3}{\log 2} \) years
• D. \( 50 \cdot \frac{\log 2}{\log 3} \) years

Hint:

For exponential growth problems, use the formula \( P(t) = P_0 e^{kt} \) and the doubling or tripling times to solve for the growth constant and time.

Answer 1

The Correct Option is B

Step 1: Use the formula for exponential growth.
The population grows exponentially, so the formula for population growth is: \[ P(t) = P_0 e^{kt} \] where \( P_0 \) is the initial population, \( k \) is the growth constant, and \( t \) is time.
Step 2: Use the doubling time.
We are told that the population doubles in 50 years, so: \[ P(50) = 2P_0 \] Using the growth formula, we get: \[ 2P_0 = P_0 e^{50k} \] Solve for \( k \) to find the growth constant.
Step 3: Use the tripling time.
To find the time for the population to triple, use: \[ 3P_0 = P_0 e^{kt} \] Solve for \( t \) to get the time for the population to triple: \[ t = 50 \cdot \frac{\log 3}{\log 2} \]
Step 4: Conclusion.
Thus, the population will become triple in \( 50 \cdot \frac{\log 3}{\log 2} \) years.