Question
Ramona's Garage charges the following labor rates. All customers are charged for at least 0.5 hr.
Ramona's Garage
Hours
0.5
1
1.5
2
2.5
Labor
$60
$90
$120
$150
$180
a. Write a linear function for the data in the table.
b. A repair job took 4 hr and 15 min and required $390 in parts. What is the total cost?
Question content area bottom
Part 1
a. A linear function for the data in the table is f(t)equals
enter your response here.
Ramona's Garage
Hours
0.5
1
1.5
2
2.5
Labor
$60
$90
$120
$150
$180
a. Write a linear function for the data in the table.
b. A repair job took 4 hr and 15 min and required $390 in parts. What is the total cost?
Question content area bottom
Part 1
a. A linear function for the data in the table is f(t)equals
enter your response here.
Answers
GPT-4o mini
To write a linear function for the data in the table, we can identify the relationship between the hours (t) and the labor cost (f(t)) based on the given information.
The data points from the table are:
- For 0.5 hours, the cost is $60.
- For 1 hour, the cost is $90.
- For 1.5 hours, the cost is $120.
- For 2 hours, the cost is $150.
- For 2.5 hours, the cost is $180.
We can observe that:
- The cost increases by $30 for each additional 0.5 hours of labor:
Now, let's calculate the slope of the linear function, which represents the rate of change in cost concerning time.
To determine the slope \( m \):
\[
m = \frac{\text{Change in cost}}{\text{Change in hours}} = \frac{90 - 60}{1 - 0.5} = \frac{30}{0.5} = 60
\]
So, the slope \( m = 60 \).
Next, we need to find the y-intercept. We can use the initial data point, (0.5, 60):
\[
f(t) = mt + b
\]
Substituting \( m = 60 \) and the point (0.5, 60):
\[
60 = 60(0.5) + b \\
60 = 30 + b \\
b = 60 - 30 = 30
\]
Now we have the linear equation:
\[
f(t) = 60t + 30
\]
However, realize that the minimum charge is at least $60 for 0.5 hr. The function can be interpreted differently between certain ranges, as there won’t be a charging for less than 0.5 hr. Hence, the pricing can be structured as a piecewise function, where the labor cost is charged for at least 0.5 hrs.
So, the cost can be formulated as:
\[
\text{If } t \text{ in hours, }
\begin{cases}
f(t) = 60t & \text{if } t \geq 0.5
\end{cases}
\]
### Summary
The linear function for the data in the table is:
\[
f(t) = 60t \quad (\text{for } t \geq 0.5)
\]
---
### Part 2
To calculate the total cost for a repair job that took 4 hours and 15 minutes:
1. Convert 4 hours and 15 minutes into hours:
- 15 minutes = 0.25 hours,
- Total time = 4 + 0.25 = 4.25 hours.
2. Calculate the labor cost using the function \( f(t) \):
\[
f(4.25) = 60 \times 4.25 = 255
\]
3. Now, add the cost of parts ($390) to the labor cost:
\[
\text{Total Cost} = \text{Labor Cost} + \text{Cost of Parts} = 255 + 390 = 645
\]
Thus, the total cost for the repair job is:
\[
\text{Total Cost} = 645
\]
Final answer:
- Total cost = $645.
The data points from the table are:
- For 0.5 hours, the cost is $60.
- For 1 hour, the cost is $90.
- For 1.5 hours, the cost is $120.
- For 2 hours, the cost is $150.
- For 2.5 hours, the cost is $180.
We can observe that:
- The cost increases by $30 for each additional 0.5 hours of labor:
Now, let's calculate the slope of the linear function, which represents the rate of change in cost concerning time.
To determine the slope \( m \):
\[
m = \frac{\text{Change in cost}}{\text{Change in hours}} = \frac{90 - 60}{1 - 0.5} = \frac{30}{0.5} = 60
\]
So, the slope \( m = 60 \).
Next, we need to find the y-intercept. We can use the initial data point, (0.5, 60):
\[
f(t) = mt + b
\]
Substituting \( m = 60 \) and the point (0.5, 60):
\[
60 = 60(0.5) + b \\
60 = 30 + b \\
b = 60 - 30 = 30
\]
Now we have the linear equation:
\[
f(t) = 60t + 30
\]
However, realize that the minimum charge is at least $60 for 0.5 hr. The function can be interpreted differently between certain ranges, as there won’t be a charging for less than 0.5 hr. Hence, the pricing can be structured as a piecewise function, where the labor cost is charged for at least 0.5 hrs.
So, the cost can be formulated as:
\[
\text{If } t \text{ in hours, }
\begin{cases}
f(t) = 60t & \text{if } t \geq 0.5
\end{cases}
\]
### Summary
The linear function for the data in the table is:
\[
f(t) = 60t \quad (\text{for } t \geq 0.5)
\]
---
### Part 2
To calculate the total cost for a repair job that took 4 hours and 15 minutes:
1. Convert 4 hours and 15 minutes into hours:
- 15 minutes = 0.25 hours,
- Total time = 4 + 0.25 = 4.25 hours.
2. Calculate the labor cost using the function \( f(t) \):
\[
f(4.25) = 60 \times 4.25 = 255
\]
3. Now, add the cost of parts ($390) to the labor cost:
\[
\text{Total Cost} = \text{Labor Cost} + \text{Cost of Parts} = 255 + 390 = 645
\]
Thus, the total cost for the repair job is:
\[
\text{Total Cost} = 645
\]
Final answer:
- Total cost = $645.