Question

Q1 There are 170 deer on a reservation. The deer population is increasing at a rate of 30% each year. The table shows the number of deer, y, after x years.

Answer 1

To model the deer population, we can use the exponential growth formula, which is given by:

\[
y = y_0 \times (1 + r)^x
\]

where:
- \( y \) is the population after \( x \) years,
- \( y_0 \) is the initial population (170 deer in this case),
- \( r \) is the growth rate (30%, or 0.30), and
- \( x \) is the number of years.

Given this information, we can write the equation for the deer population as:

\[
y = 170 \times (1 + 0.30)^x
\]

or

\[
y = 170 \times (1.30)^x
\]

1. **Calculate the deer population for specific years:**
- For \( x = 0 \) (the current year):
\[ y = 170 \times (1.30)^0 = 170 \times 1 = 170 \]
- For \( x = 1 \) (after 1 year):
\[ y = 170 \times (1.30)^1 = 170 \times 1.30 = 221 \]
- For \( x = 2 \) (after 2 years):
\[ y = 170 \times (1.30)^2 \approx 170 \times 1.69 \approx 288.3 \]
- For \( x = 3 \) (after 3 years):
\[ y = 170 \times (1.30)^3 \approx 170 \times 2.197 \approx 373.49 \]
- For \( x = 4 \) (after 4 years):
\[ y = 170 \times (1.30)^4 \approx 170 \times 2.857 \approx 485.67 \]

2. **Fill the table:**

| Years (x) | Deer Population (y) |
|-----------|----------------------|
| 0 | 170 |
| 1 | 221 |
| 2 | 288 |
| 3 | 373 |
| 4 | 486 |

You may continue this pattern to find the deer population for additional years as needed. If you need calculations for more years, let me know!