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.
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.

Answer 1

To analyze the growth of the deer population from year 1 to year 3 and from year 5 to year 10, we can use the population growth formula for exponential growth:

\[
y = P(1 + r)^t
\]

where:
- \( P \) = initial population
- \( r \) = growth rate (expressed as a decimal)
- \( t \) = number of years
- \( y \) = population after \( t \) years

For this problem:
- Initial population, \( P = 170 \)
- Growth rate, \( r = 0.30 \) (30%)

### Step 1: Calculate the population after each year up to year 10

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

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

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

Now we can summarize the populations:
- Year 1: \( 221 \)
- Year 2: \( 287.3 \)
- Year 3: \( 373.49 \)

### Step 2: Calculate the growth from year 1 to year 3

To find the growth over years 1 to 3:
- Initial Population (at year 1) = \( 221 \)
- Population at year 3 = \( 373.49 \)

The difference in population:
\[
\text{Growth} = 373.49 - 221 \approx 152.49
\]

### Step 3: Calculate the population at years 5 to 10

We continue the previous calculations:

4. **After 4 Years (x = 4)**:
\[
y = 170(1.30^4) \approx 170(2.857) \approx 485.69
\]

5. **After 5 Years (x = 5)**:
\[
y = 170(1.30^5) \approx 170(3.71293) \approx 631.19
\]

6. **After 6 Years (x = 6)**:
\[
y = 170(1.30^6) \approx 170(4.812) \approx 817.04
\]

7. **After 7 Years (x = 7)**:
\[
y = 170(1.30^7) \approx 170(6.251) \approx 1062.53
\]

8. **After 8 Years (x = 8)**:
\[
y = 170(1.30^8) \approx 170(8.168) \approx 1388.49
\]

9. **After 9 Years (x = 9)**:
\[
y = 170(1.30^9) \approx 170(10.628) \approx 1806.76
\]

10. **After 10 Years (x = 10)**:
\[
y = 170(1.30^{10}) \approx 170(13.223) \approx 2249.86
\]

### Step 4: Calculate the growth from year 5 to year 10

To find the growth over years 5 to 10:
- Initial Population (at year 5) = \( 631.19 \)
- Population at year 10 = \( 2249.86 \)

The difference in population:
\[
\text{Growth} = 2249.86 - 631.19 \approx 1618.67
\]

### Step 5: Compare the growth rates

- Growth from year 1 to year 3: \( 152.49 \)
- Growth from year 5 to year 10: \( 1618.67 \)

### Conclusion

The growth rate of deer from 1–3 years is much smaller compared to the growth rate of deer from 5–10 years. The growth over the 5 to 10 year period is substantially greater than the growth over the first 3 years, primarily due to the exponential nature of the population increase, which accelerates as the initial population grows larger.