Question
The table below shows the number of minutes a household phone is used daily based on the number of household members.
Household Members, x 1 2 3 4 5
Phone Usage in Minutes, P 33 65.2 98.4 134.6 168.8
Which equation best models this set of data?
A.
P = 33.2x + 0.8
B.
P = -33.2x2 - 0.8
C.
P = -33.2x - 0.8
D.
P = 33.2x2 + 0.8
Household Members, x 1 2 3 4 5
Phone Usage in Minutes, P 33 65.2 98.4 134.6 168.8
Which equation best models this set of data?
A.
P = 33.2x + 0.8
B.
P = -33.2x2 - 0.8
C.
P = -33.2x - 0.8
D.
P = 33.2x2 + 0.8
Answers
GPT-4o mini
To determine which equation best models the data provided, we should analyze the relationship between the number of household members (x) and the phone usage in minutes (P).
Let's look at the phone usage values for each number of household members:
1. For 1 member (x = 1), P = 33
2. For 2 members (x = 2), P = 65.2
3. For 3 members (x = 3), P = 98.4
4. For 4 members (x = 4), P = 134.6
5. For 5 members (x = 5), P = 168.8
Now, we can calculate the differences in phone usage between consecutive household members to see if the relationship is linear:
- 2 - 1 member: 65.2 - 33 = 32.2
- 3 - 2 members: 98.4 - 65.2 = 33.2
- 4 - 3 members: 134.6 - 98.4 = 36.2
- 5 - 4 members: 168.8 - 134.6 = 34.2
The increments are roughly constant, which suggests a linear relationship.
To find the slope (m), we can use the change between two points, for example between the 1-member and 2-member case:
Slope \(m\) = (65.2 - 33) / (2 - 1) = 32.2.
To express this in the equation of the line:
The initial value (when x=1) is 33 minutes.
Thus, we can propose the following linear model:
The equation in point-slope form can be written as:
\[P = mx + b\]
Where:
- m (slope) = 32.2 approximately
- b (the y-intercept) at \(x = 1\) is an adjustment.
However, the provided options do not seem to directly tell us how the intercept is formed. Let’s refine the options with knowledge of the data:
1. Each free-choice option should be linear since the data is almost linear in nature.
2. We eliminate the quadratic or complex forms since the increments appear linear.
3. Given that linear slopes seem consistent in the first two calculations, we check if the other options correspondingly match a linear behavior.
Given the context of the options, the closest linear equation represents:
**Option A:** \(P = 33.2x + 0.8\)
This format allows for a corresponding phone usage growth as household numbers increase.
### Conclusion
The best model from those provided is **Option A:** \(P = 33.2x + 0.8\).
Let's look at the phone usage values for each number of household members:
1. For 1 member (x = 1), P = 33
2. For 2 members (x = 2), P = 65.2
3. For 3 members (x = 3), P = 98.4
4. For 4 members (x = 4), P = 134.6
5. For 5 members (x = 5), P = 168.8
Now, we can calculate the differences in phone usage between consecutive household members to see if the relationship is linear:
- 2 - 1 member: 65.2 - 33 = 32.2
- 3 - 2 members: 98.4 - 65.2 = 33.2
- 4 - 3 members: 134.6 - 98.4 = 36.2
- 5 - 4 members: 168.8 - 134.6 = 34.2
The increments are roughly constant, which suggests a linear relationship.
To find the slope (m), we can use the change between two points, for example between the 1-member and 2-member case:
Slope \(m\) = (65.2 - 33) / (2 - 1) = 32.2.
To express this in the equation of the line:
The initial value (when x=1) is 33 minutes.
Thus, we can propose the following linear model:
The equation in point-slope form can be written as:
\[P = mx + b\]
Where:
- m (slope) = 32.2 approximately
- b (the y-intercept) at \(x = 1\) is an adjustment.
However, the provided options do not seem to directly tell us how the intercept is formed. Let’s refine the options with knowledge of the data:
1. Each free-choice option should be linear since the data is almost linear in nature.
2. We eliminate the quadratic or complex forms since the increments appear linear.
3. Given that linear slopes seem consistent in the first two calculations, we check if the other options correspondingly match a linear behavior.
Given the context of the options, the closest linear equation represents:
**Option A:** \(P = 33.2x + 0.8\)
This format allows for a corresponding phone usage growth as household numbers increase.
### Conclusion
The best model from those provided is **Option A:** \(P = 33.2x + 0.8\).