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.

Question

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 in a easy and simple way to write down

Answer 1

To compare the growth rate of the deer population from 1 to 3 years and from 5 to 10 years, we need to calculate the number of deer at each of these time points using the formula for exponential growth.

The formula for exponential growth is: \[ y = y_0 \times (1 + r)^x \] Where:

• \(y\) is the final population,
• \(y_0\) is the initial population (170 deer),
• \(r\) is the growth rate (30% or 0.30),
• \(x\) is the number of years.

Step 1: Calculate the deer population after 1, 2, and 3 years

1. After 1 year (x = 1): \[ y = 170 \times (1 + 0.30)^1 = 170 \times 1.30 = 221 \]

2. After 2 years (x = 2): \[ y = 170 \times (1 + 0.30)^2 = 170 \times 1.30^2 = 170 \times 1.69 = 287.3 \approx 287 \]

3. After 3 years (x = 3): \[ y = 170 \times (1 + 0.30)^3 = 170 \times 1.30^3 = 170 \times 2.197 = 373.49 \approx 373 \]

Step 2: Calculate the deer population after 5, 6, 7, 8, 9, and 10 years

1. After 5 years (x = 5): \[ y = 170 \times (1 + 0.30)^5 = 170 \times 1.30^5 = 170 \times 3.71293 \approx 630.2 \approx 630 \]

2. After 6 years (x = 6): \[ y = 170 \times (1 + 0.30)^6 = 170 \times 1.30^6 = 170 \times 4.828 \approx 820.76 \approx 821 \]

3. After 7 years (x = 7): \[ y = 170 \times (1 + 0.30)^7 = 170 \times 1.30^7 = 170 \times 6.277 \approx 1067.09 \approx 1067 \]

4. After 8 years (x = 8): \[ y = 170 \times (1 + 0.30)^8 = 170 \times 1.30^8 = 170 \times 8.164 \approx 1387.16 \approx 1387 \]

5. After 9 years (x = 9): \[ y = 170 \times (1 + 0.30)^9 = 170 \times 1.30^9 = 170 \times 10.619 \approx 1805.23 \approx 1805 \]

6. After 10 years (x = 10): \[ y = 170 \times (1 + 0.30)^{10} = 170 \times 1.30^{10} = 170 \times 13.863 \approx 2356.71 \approx 2357 \]

Step 3: Calculate the population growth

1. From year 1 to 3:

   • Starting population after 1 year: 221
   • Ending population after 3 years: 373

   Growth from year 1 to 3: \[ 373 - 221 = 152 \]

2. From year 5 to 10:

   • Starting population after 5 years: 630
   • Ending population after 10 years: 2357

   Growth from year 5 to 10: \[ 2357 - 630 = 1727 \]

Conclusion

• Growth from year 1 to 3: 152 deer
• Growth from year 5 to 10: 1727 deer

The growth rate of deer from years 5 to 10 (1727 deer) is significantly higher than the growth rate from years 1 to 3 (152 deer). This shows that the deer population grows increasingly faster as time goes on.