- Identify the linear relationship from the provided data as \(L(h) = 60h + 30\).
- Convert the repair time of 4 hours and 15 minutes to 4.25 hours.
- Calculate the labor cost for 4.25 hours as \(\$285\).
- Add the parts cost of \(\$390\) to the labor cost to find the total: \(\boxed{\$675}\).
Explanation
- Analysis of the Problem
We are given data representing labor costs at Ramona's Garage based on the time spent on a repair job. The hours worked are denoted by \(h\), and the labor cost is denoted by \(L(h)\). The provided table lists pairs of \((h, L(h))\) as \((0.5, 60), (1, 90), (1.5, 120), (2, 150),\) and \((2.5, 180)\). Our goal is to determine the linear function \(L(h) = mh + b\) that models these costs and use it to find the total cost of a repair job that takes 4 hours and 15 minutes with additional parts costing \(\$390\).
- Finding the Linear Function
To find the linear function \(L(h) = mh + b\), we first calculate the slope \(m\) using two points from the table, for instance, \((0.5, 60)\) and \((1, 90)\). The slope \(m\) is given by the change in cost divided by the change in hours:
\[m = \frac{L(h_2) - L(h_1)}{h_2 - h_1} = \frac{90 - 60}{1 - 0.5} = \frac{30}{0.5} = 60.\]
Next, we substitute the slope and one point into the equation \(L(h) = mh + b\) to find the y-intercept \(b\):
\[60 = 60(0.5) + b\]
\[60 = 30 + b\]
\[b = 30.\]
Thus, the linear function for the labor cost is \(L(h) = 60h + 30\).
- Calculating Total Cost
To calculate the total cost of a repair job that takes 4 hours and 15 minutes, we first convert the time to hours:
\[4 \text{ h and } 15 \text{ min} = 4 + \frac{15}{60} \text{ h} = 4 + 0.25 \text{ h} = 4.25 \text{ h}.\]
Now, calculate the labor cost \(L(4.25)\) using the function determined previously:
\[L(4.25) = 60(4.25) + 30 = 255 + 30 = 285.\]
The labor cost for the job is \(\$285\). Given that the parts cost is \(\$390\), the total cost is:
\[\text{Total Cost} = \text{Labor Cost} + \text{Parts Cost} = \$285 + \$390 = \$675.\]
Therefore, the total cost for the repair job is \(\$675\).
Examples
Linear modeling is essential for businesses to set transparent pricing. For instance, if you provide freelance services like graphic design or tutoring, you might charge a base fee plus an hourly rate. By defining your income as a linear function, you can quickly estimate project costs for clients, ensuring both fairness and professional clarity. This math application is fundamental in managing finances, setting budgets, and scaling service-based businesses.