Question

Sheila is a wildlife biologist. At the beginning of each year, she tracks the wild turkey and white-tail deer population in the game reserve where she works.
At the first year Sheila counted
\[12\] wild turkeys, and their number increases by approximately
\[40\%\] each year.
At the first year Sheila counted
\[18\] white-tail deer, and their number increases by
\[10\] additional deer per year.
What is the first year in which Sheila counts more turkeys than deer?

Answer 1

To determine the first year in which Sheila counts more wild turkeys than white-tail deer, we can model the populations of turkeys and deer mathematically.

**Wild Turkeys:**
- Initial population: \( T_0 = 12 \)
- Annual growth rate: \( 40\% \)

The population of wild turkeys after \( n \) years can be expressed as:
\[
T_n = T_0 \times (1 + 0.4)^n = 12 \times (1.4)^n
\]

**White-tail Deer:**
- Initial population: \( D_0 = 18 \)
- Annual growth: \( 10 \) additional deer.

The population of white-tail deer after \( n \) years can be modeled as:
\[
D_n = D_0 + 10n = 18 + 10n
\]

**Finding when turkeys exceed deer:**
We need to find the smallest integer \( n \) such that:
\[
T_n > D_n
\]
Substituting the expressions for \( T_n \) and \( D_n \):
\[
12 \times (1.4)^n > 18 + 10n
\]

Now we will evaluate this inequality for successive values of \( n \).

1. **For \( n = 1 \):**
\[
T_1 = 12 \times (1.4) = 16.8
\]
\[
D_1 = 18 + 10 \times 1 = 28
\]
\( 16.8 < 28 \)

2. **For \( n = 2 \):**
\[
T_2 = 12 \times (1.4)^2 = 12 \times 1.96 = 23.52
\]
\[
D_2 = 18 + 10 \times 2 = 38
\]
\( 23.52 < 38 \)

3. **For \( n = 3 \):**
\[
T_3 = 12 \times (1.4)^3 = 12 \times 2.744 = 32.928
\]
\[
D_3 = 18 + 10 \times 3 = 48
\]
\( 32.928 < 48 \)

4. **For \( n = 4 \):**
\[
T_4 = 12 \times (1.4)^4 = 12 \times 3.8416 \approx 46.0992
\]
\[
D_4 = 18 + 10 \times 4 = 58
\]
\( 46.0992 < 58 \)

5. **For \( n = 5 \):**
\[
T_5 = 12 \times (1.4)^5 = 12 \times 5.378 \approx 64.536
\]
\[
D_5 = 18 + 10 \times 5 = 68
\]
\( 64.536 < 68 \)

6. **For \( n = 6 \):**
\[
T_6 = 12 \times (1.4)^6 = 12 \times 7.529536 \approx 90.35443
\]
\[
D_6 = 18 + 10 \times 6 = 78
\]
\( 90.35443 > 78 \)

Thus, at \( n = 6 \) (which is the 6th year), Sheila counts more wild turkeys than white-tail deer.

**Conclusion:**
The first year in which Sheila counts more turkeys than deer is:
\[
\text{Year } 6
\]