A colony of bacteria numbers 2,000 at 1 P.M. and increases 20 percent per hour . What is the population at 4 P.M. that same day ?

Options

A. 2800
B. 3456
C. 3200
D. 3800

Answer

To find the population of the bacteria at 4 PM, we can use the formula for exponential growth. The population increases by 20% every hour, which can be expressed as a multiplication factor of 1.2 (since 100% + 20% = 120%, or 1.2 in decimal form).

1. **Initial Population**:
- At 1 PM, the initial population \( P_0 = 2000 \).

2. **Growth Rate**:
- The growth rate per hour is 20%, which can be expressed as a multiplication factor of \( 1.2 \).

3. **Time Duration**:
- From 1 PM to 4 PM, there are 3 hours (1 PM to 2 PM, 2 PM to 3 PM, and 3 PM to 4 PM).

4. **Population Calculation**:
- The population at 4 PM can be calculated using the formula:
\[ P = P_0 \times (growth\ factor)^{number\ of\ hours} \]
- Here, \( P_0 = 2000 \), the growth factor is \( 1.2 \), and the number of hours is 3.
\[ P = 2000 \times (1.2)^3 \]

5. **Calculating \( (1.2)^3 \)**:
- First, calculate \( (1.2)^2 \):
\[ (1.2)^2 = 1.2 \times 1.2 = 1.44 \]
- Now multiply by \( 1.2 \) again to find \( (1.2)^3 \):
\[ (1.2)^3 = 1.44 \times 1.2 = 1.728 \]

6. **Final Calculation**:
- Now substitute back into the population formula:
\[ P = 2000 \times 1.728 = 3456 \]

7. **Conclusion**:
- Therefore, the population of the bacteria at 4 PM is \( 3456 \).