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.

Use the table to answer the question.

Years (x) 0 1 3 5 10
Deer (y) 170 221 373 631 2344
How does the growth rate of deer from 1–3 years compare to the growth rate of deer from 5– 10 years? Show your work.

(2 points)

1 answer

To compare the growth rate of deer from 1 to 3 years and from 5 to 10 years, we can calculate the percentage increases in the deer population over those specific time intervals.

1. Calculate growth from year 1 to year 3:

  • At year 1, the number of deer \( y_1 = 221 \)
  • At year 3, the number of deer \( y_3 = 373 \)

The increase in deer from year 1 to year 3 is:

\[ y_3 - y_1 = 373 - 221 = 152 \]

The percentage increase is calculated as follows:

\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Initial Value}} \right) \times 100 \]

Substituting values:

\[ \text{Percentage Increase} = \left( \frac{152}{221} \right) \times 100 \approx 68.87% \]

2. Calculate growth from year 5 to year 10:

  • At year 5, the number of deer \( y_5 = 631 \)
  • At year 10, the number of deer \( y_{10} = 2344 \)

The increase in deer from year 5 to year 10 is:

\[ y_{10} - y_5 = 2344 - 631 = 1713 \]

The percentage increase is calculated as follows:

\[ \text{Percentage Increase} = \left( \frac{1713}{631} \right) \times 100 \approx 271.56% \]

Comparison of Growth Rates:

  • The growth rate from year 1 to year 3 is approximately 68.87%.
  • The growth rate from year 5 to year 10 is approximately 271.56%.

Conclusion: The growth rate of deer from year 5 to year 10 (approximately 271.56%) is significantly higher than the growth rate from year 1 to year 3 (approximately 68.87%). Thus, the population of deer is experiencing much more rapid growth in the later years compared to the earlier years.