Question
ack works after school. Each day he earns a set amount, plus an hourly wage.
Hours 1 1.5 2 2.5 3
Pay 18 23 28 33 38
Part A: Write a linear function f Jack can use to determine his pay
f(x) = 10x+8
f(x) = 15x+3
f(x) = 0.5x+1
f(x) = x+18
Part B: Which is a reasonable domain?
0≤x≤6
0≤x≤24
0≤x≤68
0≤x≤248
Assume Jack works from 2:30pm to 7:00pm. How much would he earn? (hint: How many hours did he work?)
$33
$45
$35.50
$53
Hours 1 1.5 2 2.5 3
Pay 18 23 28 33 38
Part A: Write a linear function f Jack can use to determine his pay
f(x) = 10x+8
f(x) = 15x+3
f(x) = 0.5x+1
f(x) = x+18
Part B: Which is a reasonable domain?
0≤x≤6
0≤x≤24
0≤x≤68
0≤x≤248
Assume Jack works from 2:30pm to 7:00pm. How much would he earn? (hint: How many hours did he work?)
$33
$45
$35.50
$53
Answers
GPT-4o mini
To find the linear function that describes Jack's pay based on the hours he works, we can first analyze the given data.
### Part A: Writing a linear function
Let’s look at the provided hours and corresponding pay:
- For 1 hour, he gets $18
- For 1.5 hours, he gets $23
- For 2 hours, he gets $28
- For 2.5 hours, he gets $33
- For 3 hours, he gets $38
We can see that as he works more hours, he earns more money in a linear fashion.
Calculating the differences in pay for each additional hour worked:
- From 1 hour to 1.5 hours: $23 - $18 = $5 (increase of 5 for 0.5 hours)
- From 1.5 to 2 hours: $28 - $23 = $5
- From 2 to 2.5 hours: $33 - $28 = $5
- From 2.5 to 3 hours: $38 - $33 = $5
This indicates he earns $10 per hour (since $5 per 0.5 hours translates to $10 per hour).
Let's determine the fixed base pay (which is the pay for 0 hours worked).
Looking at the pay at 0 hours, we find he would earn $18 for the first hour, suggesting that the linear equation can be modeled as:
- Initial pay = $18
- Cost per hour = $10
So, the equation is:
\[
f(x) = 10x + 8
\]
Thus:
- The correct answer for Part A is: **f(x) = 10x + 8**
### Part B: Reasonable domain
For a reasonable domain of hours worked, we know Jack generally works a few hours after school.
Looking at the options:
- 0 ≤ x ≤ 6: This makes sense as it covers a reasonable amount of after school work.
- 0 ≤ x ≤ 24: Too broad as Jack is not likely working for a full day each day.
- 0 ≤ x ≤ 68: Unlikely for regular after school shifts.
- 0 ≤ x ≤ 248: Not reasonable at all.
The reasonable domain is: **0 ≤ x ≤ 6**
### Earnings calculation based on work hours
Assuming Jack works from 2:30 PM to 7:00 PM:
- Total work hours = 4.5 hours (from 2:30 PM to 7:00 PM)
Using the function f(x) = 10x + 8:
\[
f(4.5) = 10(4.5) + 8 = 45 + 8 = 53
\]
Thus, Jack would earn:
- **$53**
### Summary of Answers
- Part A: **f(x) = 10x + 8**
- Part B: **0 ≤ x ≤ 6**
- Earnings for 4.5 hours: **$53**
### Part A: Writing a linear function
Let’s look at the provided hours and corresponding pay:
- For 1 hour, he gets $18
- For 1.5 hours, he gets $23
- For 2 hours, he gets $28
- For 2.5 hours, he gets $33
- For 3 hours, he gets $38
We can see that as he works more hours, he earns more money in a linear fashion.
Calculating the differences in pay for each additional hour worked:
- From 1 hour to 1.5 hours: $23 - $18 = $5 (increase of 5 for 0.5 hours)
- From 1.5 to 2 hours: $28 - $23 = $5
- From 2 to 2.5 hours: $33 - $28 = $5
- From 2.5 to 3 hours: $38 - $33 = $5
This indicates he earns $10 per hour (since $5 per 0.5 hours translates to $10 per hour).
Let's determine the fixed base pay (which is the pay for 0 hours worked).
Looking at the pay at 0 hours, we find he would earn $18 for the first hour, suggesting that the linear equation can be modeled as:
- Initial pay = $18
- Cost per hour = $10
So, the equation is:
\[
f(x) = 10x + 8
\]
Thus:
- The correct answer for Part A is: **f(x) = 10x + 8**
### Part B: Reasonable domain
For a reasonable domain of hours worked, we know Jack generally works a few hours after school.
Looking at the options:
- 0 ≤ x ≤ 6: This makes sense as it covers a reasonable amount of after school work.
- 0 ≤ x ≤ 24: Too broad as Jack is not likely working for a full day each day.
- 0 ≤ x ≤ 68: Unlikely for regular after school shifts.
- 0 ≤ x ≤ 248: Not reasonable at all.
The reasonable domain is: **0 ≤ x ≤ 6**
### Earnings calculation based on work hours
Assuming Jack works from 2:30 PM to 7:00 PM:
- Total work hours = 4.5 hours (from 2:30 PM to 7:00 PM)
Using the function f(x) = 10x + 8:
\[
f(4.5) = 10(4.5) + 8 = 45 + 8 = 53
\]
Thus, Jack would earn:
- **$53**
### Summary of Answers
- Part A: **f(x) = 10x + 8**
- Part B: **0 ≤ x ≤ 6**
- Earnings for 4.5 hours: **$53**