A certain population of insects starts at 16 and doubles every 6 days. What is the population after 60 days?

Options

A. `2^(6)`
B. `2^(10)`
C. `2^(14)`
D. `2^(32)`

Answer

To solve the problem of determining the population of insects after 60 days, we can follow these steps:

### Step 1: Identify the initial population and the doubling period
- The initial population of insects is 16.
- The population doubles every 6 days.

### Step 2: Calculate the number of doubling periods in 60 days
- To find out how many times the population doubles in 60 days, we divide the total time (60 days) by the doubling period (6 days):
\[ \text{Number of doublings} = \frac{60 \text{ days}}{6 \text{ days}} = 10 \]

### Step 3: Determine the population after 10 doublings
- The population after each doubling can be expressed as:
\[ \text{Population after } n \text{ doublings} = \text{Initial population} \times 2^n \]
- Substituting the values we have:
\[ \text{Population after 10 doublings} = 16 \times 2^{10} \]

### Step 4: Simplify the expression
- We can express 16 as a power of 2:
\[ 16 = 2^4 \]
- Therefore, we can rewrite the population formula:
\[ \text{Population after 10 doublings} = 2^4 \times 2^{10} = 2^{4 + 10} = 2^{14} \]

### Step 5: Calculate the final population
- Now, we need to calculate \(2^{14}\):
\[ 2^{14} = 16384 \]
- Thus, the population of insects after 60 days is **16384**.

### Final Answer
The population after 60 days is **16384**.

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