A population, T(x), of wild turkeys, in a certain rural area is represented by the function, `T(x)=17(1.15)^(2x)`, where x is number of years since 2010. According to this model, how many more turkeys are in the population for the year 2015 than were avaible for 2010?

Options

A. `46`
B. `49`
C. `51`
D. `68`

Answer

To find out how many more turkeys are in the population for the year 2015 than were available in 2010, we can follow these steps:

### Step 1: Determine the value of x for the years 2010 and 2015.
- For the year 2010, \( x = 0 \) (since it is the starting point).
- For the year 2015, \( x = 5 \) (since 2015 is 5 years after 2010).

### Step 2: Calculate the population of turkeys in 2010.
Using the function \( T(x) = 17(1.15)^{2x} \):
- Substitute \( x = 0 \):
\[ T(0) = 17(1.15)^{2 \cdot 0} = 17(1) = 17 \]
So, the population in 2010 is 17 turkeys.

### Step 3: Calculate the population of turkeys in 2015.
Using the same function:
- Substitute \( x = 5 \):
\[ T(5) = 17(1.15)^{2 \cdot 5} = 17(1.15)^{10} \]

### Step 4: Calculate \( (1.15)^{10} \).
Using a calculator or exponentiation:
\[ (1.15)^{10} \approx 4.045557 \]
Now, substitute this value back into the equation:
\[ T(5) = 17 \times 4.045557 \approx 68.774449 \]
So, the population in 2015 is approximately 68.77 turkeys.

### Step 5: Find the difference in population between 2015 and 2010.
Now, subtract the population in 2010 from the population in 2015:
\[ \text{Difference} = T(5) - T(0) \approx 68.77 - 17 \approx 51.77 \]
Rounding this to the nearest whole number gives us approximately 52 turkeys.

### Step 6: Conclusion
Thus, the number of more turkeys in 2015 than in 2010 is approximately 52.

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