Question
The population of deer in a park doubles every 10 years. The growth is modeled by the equation y=100(2)^x
where y
is the number of deer and x
is the number of ten-year periods.
Which data point represents a solution and interpretation to the equation?
Group of answer choices
The data point (10,200) represents 200 deer after 100 years.
The data point (2,400) represents 400 deer after 2 years.
The data point (10,200) represents 200 deer after 10 years.
The data point (2,400) represents 400 deer after 20 years.
where y
is the number of deer and x
is the number of ten-year periods.
Which data point represents a solution and interpretation to the equation?
Group of answer choices
The data point (10,200) represents 200 deer after 100 years.
The data point (2,400) represents 400 deer after 2 years.
The data point (10,200) represents 200 deer after 10 years.
The data point (2,400) represents 400 deer after 20 years.
Answers
Answered by
GPT-4o mini
To analyze the equation \( y = 100(2)^x \), we need to understand what the variables represent:
- \( y \) is the number of deer.
- \( x \) is the number of ten-year periods.
Given this, let's evaluate the provided data points:
1. **The data point (10, 200) represents 200 deer after 100 years.**
- **Interpretation:** \( x = 10 \) means 10 ten-year periods, which is 100 years.
- Calculate \( y \): \( y = 100(2)^{10} = 100 \times 1024 = 102400 \). This point is incorrect.
2. **The data point (2, 400) represents 400 deer after 2 years.**
- **Interpretation:** \( x = 2 \) means 2 ten-year periods, which is 20 years (not 2 years).
- Calculate \( y \): \( y = 100(2)^2 = 100 \times 4 = 400 \). This represents 400 deer after 20 years, so this part is correct. However, the time interpretation is incorrect.
3. **The data point (10, 200) represents 200 deer after 10 years.**
- **Interpretation:** \( x = 10 \) means 10 ten-year periods, which is 100 years, not 10 years. This point is incorrect.
4. **The data point (2, 400) represents 400 deer after 20 years.**
- **Interpretation:** As calculated previously, after 2 ten-year periods (20 years), the population would indeed be 400 deer. This interpretation is correct.
Based on the evaluations, the correct data point is:
- **The data point (2, 400) represents 400 deer after 20 years.**
- \( y \) is the number of deer.
- \( x \) is the number of ten-year periods.
Given this, let's evaluate the provided data points:
1. **The data point (10, 200) represents 200 deer after 100 years.**
- **Interpretation:** \( x = 10 \) means 10 ten-year periods, which is 100 years.
- Calculate \( y \): \( y = 100(2)^{10} = 100 \times 1024 = 102400 \). This point is incorrect.
2. **The data point (2, 400) represents 400 deer after 2 years.**
- **Interpretation:** \( x = 2 \) means 2 ten-year periods, which is 20 years (not 2 years).
- Calculate \( y \): \( y = 100(2)^2 = 100 \times 4 = 400 \). This represents 400 deer after 20 years, so this part is correct. However, the time interpretation is incorrect.
3. **The data point (10, 200) represents 200 deer after 10 years.**
- **Interpretation:** \( x = 10 \) means 10 ten-year periods, which is 100 years, not 10 years. This point is incorrect.
4. **The data point (2, 400) represents 400 deer after 20 years.**
- **Interpretation:** As calculated previously, after 2 ten-year periods (20 years), the population would indeed be 400 deer. This interpretation is correct.
Based on the evaluations, the correct data point is:
- **The data point (2, 400) represents 400 deer after 20 years.**