Question
Which of the following situations could describe the graph?
Juveldy puts $20 in a high-yield savings account. His investment triples every year.
Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.
Niall rents a bike for $20 per hour.
There are 200 cockroaches in a building, and the population triples every day.
Juveldy puts $20 in a high-yield savings account. His investment triples every year.
Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.
Niall rents a bike for $20 per hour.
There are 200 cockroaches in a building, and the population triples every day.
Answers
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GPT-4o mini
To determine which situation corresponds to a specific type of graph, let's analyze each option:
1. **Juveldy puts $20 in a high-yield savings account. His investment triples every year.**
This situation describes exponential growth because the investment triples, which is a multiplication by a constant factor (3). The value of the investment can be expressed as \( V(t) = 20 \cdot 3^t \), where \( t \) is the number of years. This graph would show a rapidly increasing curve (exponential growth).
2. **Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.**
This situation also describes exponential growth, but at a different rate. The value can be expressed as \( V(t) = 20 \cdot (1.2)^t \), where \( t \) is the number of years. The graph would show an increasing curve, but not as steep as Juveldy's investment.
3. **Niall rents a bike for $20 per hour.**
This situation represents a linear relationship, as the cost increases by a fixed amount ($20) for each hour that passes. The graph of this scenario would be a straight line with a slope corresponding to the hourly rate.
4. **There are 200 cockroaches in a building, and the population triples every day.**
Similar to Juveldy's investment, this scenario also describes exponential growth because the population triples daily. The value can be expressed as \( P(t) = 200 \cdot 3^t \), where \( t \) is the number of days. This graph would show a steep upward curve.
### Summary:
- **Exponential growth (tripling)**: Juveldy and the cockroach population.
- **Exponential growth (20% increase)**: Sienna.
- **Linear growth (fixed rate)**: Niall.
If you're attempting to identify a specific graph, you should consider the characteristics of each situation described, focusing particularly on whether they illustrate exponential growth (curved graph) or linear growth (straight line). If you have a specific graph to analyze, compare its shape with these descriptions.
1. **Juveldy puts $20 in a high-yield savings account. His investment triples every year.**
This situation describes exponential growth because the investment triples, which is a multiplication by a constant factor (3). The value of the investment can be expressed as \( V(t) = 20 \cdot 3^t \), where \( t \) is the number of years. This graph would show a rapidly increasing curve (exponential growth).
2. **Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.**
This situation also describes exponential growth, but at a different rate. The value can be expressed as \( V(t) = 20 \cdot (1.2)^t \), where \( t \) is the number of years. The graph would show an increasing curve, but not as steep as Juveldy's investment.
3. **Niall rents a bike for $20 per hour.**
This situation represents a linear relationship, as the cost increases by a fixed amount ($20) for each hour that passes. The graph of this scenario would be a straight line with a slope corresponding to the hourly rate.
4. **There are 200 cockroaches in a building, and the population triples every day.**
Similar to Juveldy's investment, this scenario also describes exponential growth because the population triples daily. The value can be expressed as \( P(t) = 200 \cdot 3^t \), where \( t \) is the number of days. This graph would show a steep upward curve.
### Summary:
- **Exponential growth (tripling)**: Juveldy and the cockroach population.
- **Exponential growth (20% increase)**: Sienna.
- **Linear growth (fixed rate)**: Niall.
If you're attempting to identify a specific graph, you should consider the characteristics of each situation described, focusing particularly on whether they illustrate exponential growth (curved graph) or linear growth (straight line). If you have a specific graph to analyze, compare its shape with these descriptions.
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