Question
Create a sequence to show exponential growth where the population of 120 is doubling every day.(1 point)
Responses
120, 240, 480 . . .
120, 240, 480 . . .
120, 360, 1,080 . . .
120, 360, 1,080 . . .
120, 60, 30 . . .
120, 60, 30 . . .
120, 240, 480, 360, 480 . . .
Responses
120, 240, 480 . . .
120, 240, 480 . . .
120, 360, 1,080 . . .
120, 360, 1,080 . . .
120, 60, 30 . . .
120, 60, 30 . . .
120, 240, 480, 360, 480 . . .
Answers
GPT-4o mini
The correct sequence to show exponential growth where the population of 120 is doubling every day would be:
**120, 240, 480, 960, 1920 . . .**
However, from the provided options, the closest correct sequence is:
**120, 240, 480 . . .**
This option shows the population doubling as expected.
**120, 240, 480, 960, 1920 . . .**
However, from the provided options, the closest correct sequence is:
**120, 240, 480 . . .**
This option shows the population doubling as expected.
adaleta
Select the response that creates an accurate formula for the sequence 2, 3.5, 6.125. 10.71875 . . .
.(1 point)
Responses
y=21.5x
y equals Start Fraction 2 over 1.5 x End Fraction
y=2(1.75)x−1
y equals 2 left parenthesis 1.75 right parenthesis superscript x baseline
y=2+1.75x
y equals 2 plus 1.75 x
y=2(1.5)x
.(1 point)
Responses
y=21.5x
y equals Start Fraction 2 over 1.5 x End Fraction
y=2(1.75)x−1
y equals 2 left parenthesis 1.75 right parenthesis superscript x baseline
y=2+1.75x
y equals 2 plus 1.75 x
y=2(1.5)x
GPT-4o mini
To find an accurate formula for the sequence \(2, 3.5, 6.125, 10.71875, \ldots\), we can analyze the pattern.
It seems that each term is multiplied by a common ratio which appears to be \(1.75\). Let's check this by calculating the next term based on the previous one:
- \(2 \times 1.75 = 3.5\)
- \(3.5 \times 1.75 = 6.125\)
- \(6.125 \times 1.75 = 10.71875\)
Given this pattern, the formula that fits this sequence is:
\[ y = 2(1.75)^{x-1} \]
Thus, the correct response is:
**y = 2(1.75)^{x-1}**
It seems that each term is multiplied by a common ratio which appears to be \(1.75\). Let's check this by calculating the next term based on the previous one:
- \(2 \times 1.75 = 3.5\)
- \(3.5 \times 1.75 = 6.125\)
- \(6.125 \times 1.75 = 10.71875\)
Given this pattern, the formula that fits this sequence is:
\[ y = 2(1.75)^{x-1} \]
Thus, the correct response is:
**y = 2(1.75)^{x-1}**
adaleta
Use the image to answer the question.
A concave-up, increasing curve is plotted on a coordinate plane. The x-axis ranges from 0 to 4 in increments of 0.1 but labeled at 0.5 intervals. The y-axis ranges from 0 to 800 in increments of 50 but labeled at intervals of 200. The curve passes through 4 marked and labeled points.
Which of the following situations could describe the graph?
(1 point)
Responses
There are 200 cockroaches in a building, and the population triples every day.
There are 200 cockroaches in a building, and the population triples every day.
Niall rents a bike for $20 per hour.
Niall rents a bike for $20 per hour.
Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.
Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.
Juveldy puts $20 in a high-yield savings account. His investment triples every year.
A concave-up, increasing curve is plotted on a coordinate plane. The x-axis ranges from 0 to 4 in increments of 0.1 but labeled at 0.5 intervals. The y-axis ranges from 0 to 800 in increments of 50 but labeled at intervals of 200. The curve passes through 4 marked and labeled points.
Which of the following situations could describe the graph?
(1 point)
Responses
There are 200 cockroaches in a building, and the population triples every day.
There are 200 cockroaches in a building, and the population triples every day.
Niall rents a bike for $20 per hour.
Niall rents a bike for $20 per hour.
Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.
Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.
Juveldy puts $20 in a high-yield savings account. His investment triples every year.
GPT-4o mini
To determine which situation could describe the concave-up, increasing curve you described, we need to consider the nature of each scenario.
1. **There are 200 cockroaches in a building, and the population triples every day.**
- This represents exponential growth and would result in a rapidly increasing curve that becomes very steep, likely leading to a curve that becomes concave down after a few days due to the nature of exponential growth.
2. **Niall rents a bike for $20 per hour.**
- This situation results in a linear increase, as the cost increases by a constant rate every hour. The curve would not be concave up.
3. **Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.**
- This represents exponential growth as well (20% increase annually), resulting in a curve that could be concave up over time but will increase at a slower rate compared to situations with tripling.
4. **Juveldy puts $20 in a high-yield savings account. His investment triples every year.**
- Similar to the cockroach population, tripling each year would lead to exponential growth and a curve that is steep and increases rapidly, likely also becoming concave down.
Given your description of the curve being concave up and increasing, the best response that fits is:
**Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.**
This scenario describes an exponential growth that stays concave up as it appreciates by a consistent percentage each year.
1. **There are 200 cockroaches in a building, and the population triples every day.**
- This represents exponential growth and would result in a rapidly increasing curve that becomes very steep, likely leading to a curve that becomes concave down after a few days due to the nature of exponential growth.
2. **Niall rents a bike for $20 per hour.**
- This situation results in a linear increase, as the cost increases by a constant rate every hour. The curve would not be concave up.
3. **Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.**
- This represents exponential growth as well (20% increase annually), resulting in a curve that could be concave up over time but will increase at a slower rate compared to situations with tripling.
4. **Juveldy puts $20 in a high-yield savings account. His investment triples every year.**
- Similar to the cockroach population, tripling each year would lead to exponential growth and a curve that is steep and increases rapidly, likely also becoming concave down.
Given your description of the curve being concave up and increasing, the best response that fits is:
**Sienna buys a vintage record for $20. Its value appreciates by 20 percent each year.**
This scenario describes an exponential growth that stays concave up as it appreciates by a consistent percentage each year.
adaleta
Lori buys five goldfish for her tank. Her research shows that they will reproduce at a rate of 32 percent each month. Which statement discusses a key feature of the graph of this situation?(1 point)
Responses
The points on the graph will have a common ratio of 1.32.
The points on the graph will have a common ratio of 1.32.
The graph will begin with the point (1,5)
because Lori starts with five fish.
The graph will begin with the point left parenthesis 1 comma 5 right parenthesis because Lori starts with five fish.
The graph will have a constant rate of change.
The graph will have a constant rate of change.
The points on the graph will have a common ratio of 32.
Responses
The points on the graph will have a common ratio of 1.32.
The points on the graph will have a common ratio of 1.32.
The graph will begin with the point (1,5)
because Lori starts with five fish.
The graph will begin with the point left parenthesis 1 comma 5 right parenthesis because Lori starts with five fish.
The graph will have a constant rate of change.
The graph will have a constant rate of change.
The points on the graph will have a common ratio of 32.
GPT-4o mini
To analyze the situation where Lori has five goldfish that reproduce at a rate of 32 percent each month, we can break down the key features of the graph of this scenario:
1. **Common Ratio:** Since the goldfish reproduce at a rate of 32 percent, the growth factor each month for the goldfish population can be calculated as \(1 + 0.32 = 1.32\). This indicates that the population will grow to 132% of its current size each month.
2. **Starting Point:** The initial number of goldfish is 5, which means the graph will start at the point representing this initial quantity of fish.
3. **Constant Rate of Change:** Since the population grows by a percentage (32%) each month, the rate of change is not constant; it increases as the population grows.
4. **Common Ratio of 32:** This statement is incorrect because the growth factor pertains to the 32% increase, not a common ratio of 32.
Based on this analysis, the correct statement that discusses a key feature of the graph of this situation is:
**The points on the graph will have a common ratio of 1.32.**
1. **Common Ratio:** Since the goldfish reproduce at a rate of 32 percent, the growth factor each month for the goldfish population can be calculated as \(1 + 0.32 = 1.32\). This indicates that the population will grow to 132% of its current size each month.
2. **Starting Point:** The initial number of goldfish is 5, which means the graph will start at the point representing this initial quantity of fish.
3. **Constant Rate of Change:** Since the population grows by a percentage (32%) each month, the rate of change is not constant; it increases as the population grows.
4. **Common Ratio of 32:** This statement is incorrect because the growth factor pertains to the 32% increase, not a common ratio of 32.
Based on this analysis, the correct statement that discusses a key feature of the graph of this situation is:
**The points on the graph will have a common ratio of 1.32.**
adaleta
Use the image to answer the question.
A coordinate plane's axes range from 0 to 8 in 0.5-unit increments, but labeled at every 2-unit interval. A curve is plotted passing through four marked points.
What is the initial value of the sequence?
(1 point)
Responses
0
0
2
2
0.5
0.5
1
A coordinate plane's axes range from 0 to 8 in 0.5-unit increments, but labeled at every 2-unit interval. A curve is plotted passing through four marked points.
What is the initial value of the sequence?
(1 point)
Responses
0
0
2
2
0.5
0.5
1
GPT-4o mini
To determine the initial value of the sequence from the described coordinate plane, we need to look for the point on the graph where the curve starts, specifically along the y-axis (which typically represents the initial value when x = 0).
Since the axes range from 0 to 8 and are labeled at every 2-unit interval, the initial value corresponds to the y-coordinate where the curve intersects the y-axis.
Given your choices:
1. 0
2. 2
3. 0.5
4. 1
Without seeing the image, I cannot definitively identify which of these values is the initial value. However, if the curve starts at the y-axis (x = 0), you would check the point where it intersects to identify the correct response.
If you can determine where the curve intersects the y-axis or any marked points corresponding to the x-value of 0, that would give you the initial value. Based on typical behavior in these contexts, the initial value may often be small; you'll need to refer to the specific graph you have to make the final selection.
If the curve starts at 0 on the y-axis, then the answer would be **0**. If it starts at another value like 2 or 1, select accordingly based on what you observe. If you have more information or a description of the curve's start, please share that for a more precise answer!
Since the axes range from 0 to 8 and are labeled at every 2-unit interval, the initial value corresponds to the y-coordinate where the curve intersects the y-axis.
Given your choices:
1. 0
2. 2
3. 0.5
4. 1
Without seeing the image, I cannot definitively identify which of these values is the initial value. However, if the curve starts at the y-axis (x = 0), you would check the point where it intersects to identify the correct response.
If you can determine where the curve intersects the y-axis or any marked points corresponding to the x-value of 0, that would give you the initial value. Based on typical behavior in these contexts, the initial value may often be small; you'll need to refer to the specific graph you have to make the final selection.
If the curve starts at 0 on the y-axis, then the answer would be **0**. If it starts at another value like 2 or 1, select accordingly based on what you observe. If you have more information or a description of the curve's start, please share that for a more precise answer!