Question
Use the table to answer the question.
0 350
1
2
3
4
5 A certain population of beetles is increasing at a rate of 22% per month. If there are currently 350 beetles, the equation f(x)=350(1.22)^x can be used to model the situation. Finish the chart and plot the points to determine which graph correctly displays the situation.(1 point)
Responses
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.
An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 2,000 in increments of 200. The y-axis for Beetles ranges from 0 to 10 in increments of 1.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 2,000 in increments of 200. The y-axis for Beetles ranges from 0 to 10 in increments of 1.
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
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page 16 of 16
0 350
1
2
3
4
5 A certain population of beetles is increasing at a rate of 22% per month. If there are currently 350 beetles, the equation f(x)=350(1.22)^x can be used to model the situation. Finish the chart and plot the points to determine which graph correctly displays the situation.(1 point)
Responses
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.
An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 2,000 in increments of 200. The y-axis for Beetles ranges from 0 to 10 in increments of 1.
Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 2,000 in increments of 200. The y-axis for Beetles ranges from 0 to 10 in increments of 1.
A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.
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page 16 of 16
Answers
Answer
which option is it? The first second third or fourth?
Answered by
GPT-4o mini
To complete the problem, we need to evaluate the function \( f(x) = 350(1.22)^x \) to determine the number of beetles after a certain number of months (0 to 5) and fill in the table accordingly:
1. **For \( x = 0 \):**
\[
f(0) = 350(1.22)^0 = 350 \cdot 1 = 350
\]
2. **For \( x = 1 \):**
\[
f(1) = 350(1.22)^1 = 350 \cdot 1.22 = 427
\]
3. **For \( x = 2 \):**
\[
f(2) = 350(1.22)^2 = 350 \cdot 1.4884 \approx 520.94 \approx 521
\]
4. **For \( x = 3 \):**
\[
f(3) = 350(1.22)^3 = 350 \cdot 1.80489 \approx 631.71 \approx 632
\]
5. **For \( x = 4 \):**
\[
f(4) = 350(1.22)^4 = 350 \cdot 2.204118 \approx 771.44 \approx 771
\]
6. **For \( x = 5 \):**
\[
f(5) = 350(1.22)^5 = 350 \cdot 2.677551 \approx 935.14 \approx 935
\]
Now, let's summarize the completed data in the table:
```
x (months) | Number of Beetles
--------------|--------------------
0 | 350
1 | 427
2 | 521
3 | 632
4 | 771
5 | 935
```
### Analysis of the Graphs:
Given that the function models a population growth at a rate of 22% per month, the graph should display an increasing curve, indicating that the population of beetles is growing over time.
Therefore, among the options provided, the graph that displays an increasing curve (with an arrow on the top) is the correct representation of this data.
The first option you mentioned (a decreasing curve) does **not** match the growth of the beetles population.
Thus, the correct choice would be:
**An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.**
This choice displays an accurate representation of the exponential growth seen in the population of the beetles.
1. **For \( x = 0 \):**
\[
f(0) = 350(1.22)^0 = 350 \cdot 1 = 350
\]
2. **For \( x = 1 \):**
\[
f(1) = 350(1.22)^1 = 350 \cdot 1.22 = 427
\]
3. **For \( x = 2 \):**
\[
f(2) = 350(1.22)^2 = 350 \cdot 1.4884 \approx 520.94 \approx 521
\]
4. **For \( x = 3 \):**
\[
f(3) = 350(1.22)^3 = 350 \cdot 1.80489 \approx 631.71 \approx 632
\]
5. **For \( x = 4 \):**
\[
f(4) = 350(1.22)^4 = 350 \cdot 2.204118 \approx 771.44 \approx 771
\]
6. **For \( x = 5 \):**
\[
f(5) = 350(1.22)^5 = 350 \cdot 2.677551 \approx 935.14 \approx 935
\]
Now, let's summarize the completed data in the table:
```
x (months) | Number of Beetles
--------------|--------------------
0 | 350
1 | 427
2 | 521
3 | 632
4 | 771
5 | 935
```
### Analysis of the Graphs:
Given that the function models a population growth at a rate of 22% per month, the graph should display an increasing curve, indicating that the population of beetles is growing over time.
Therefore, among the options provided, the graph that displays an increasing curve (with an arrow on the top) is the correct representation of this data.
The first option you mentioned (a decreasing curve) does **not** match the growth of the beetles population.
Thus, the correct choice would be:
**An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.**
This choice displays an accurate representation of the exponential growth seen in the population of the beetles.
Answered by
GPT-4o mini
Based on the analysis, the correct option is the **second** one:
**"An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250."**
This option accurately reflects the exponential growth of the beetle population as modeled by the equation \( f(x) = 350(1.22)^x \).
**"An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250."**
This option accurately reflects the exponential growth of the beetle population as modeled by the equation \( f(x) = 350(1.22)^x \).
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