A population of beetles doubles every week. If there are 50 beetles to begin with, how many beetles will there be in 5 weeks?
Understand the Problem
The question describes a scenario where a beetle population doubles every week, starting with an initial population of 50. The goal is to determine the size of the beetle population after 5 weeks, given this exponential growth pattern.
Answer
1600
Answer for screen readers
1600
Steps to Solve
- Determine the growth factor
Since the beetle population doubles every week, the growth factor is 2.
- Determine the number of growth periods
The population grows for 5 weeks, so there are 5 growth periods.
- Calculate the population after 5 weeks
To find the population after 5 weeks, we multiply the initial population by the growth factor raised to the power of the number of growth periods:
$$ \text{Final Population} = \text{Initial Population} \times (\text{Growth Factor})^{\text{Number of Growth Periods}} $$
$$ \text{Final Population} = 50 \times (2)^5 $$
- Simplify the exponent
$$ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 $$
- Multiply to find the final population
$$ \text{Final Population} = 50 \times 32 $$
$$ \text{Final Population} = 1600 $$
1600
More Information
The beetle population grows exponentially, meaning it increases rapidly over time. Starting with just 50 beetles, the population reaches 1600 in only 5 weeks due to the doubling effect each week.
Tips
A common mistake is to multiply the initial population by the number of weeks and the growth factor, instead of using the exponential growth formula. For example, calculating $50 \times 2 \times 5 = 500$ is incorrect. It's crucial to remember that the growth is exponential, requiring the exponentiation of the growth factor.
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