a Create two expressions for the total costs and set them equal to each other.
B
b When is it more expensive to have repairs done at the dealership? At the local mechanic?
A
a 2.1 hours. See solution.
B
b The dealership costs less at first, the local mechanic costs less after 2.1 hours.
Practice makes perfect
a The total amount we will pay for car repairs at a dealership and a mechanic has two components.
The cost of the parts, which is a flat fee.
The hourly labor cost charged for the duration of the repair process.
We can model these costs with an algebraic expression. Let's write the expressions for the total cost for each location separately.
Dealership
The cost of the parts is given to be $24 and the hourly labor cost is $99. If we let the number of hours worked be x, the total labor cost charged is 99x. When writing the expression representing the cost, we need to remember to include the parts.
99x+ 24
Local Mechanic
The cost of the parts is given to be $45 and the hourly labor cost is $89. If we let the number of hours worked be x, then we can write the total cost expression.
89x+ 45
When Are the Costs Equal?
To determine the number of hours worked when we will have paid the same total amount, we need to know for what x these two expressions are equal. We can do that by equating them and solving for x.
After 2.1 hours the total costs will be identical.
b To find when the repair at the dealership costs less, we can think of both expressions representing the cost as functions of time x, where each can be graphed on a coordinate plane.
y= 99x+24
y= 89x+45
Since 2.1 hours is the point of intersection, we can find out when the repairs will cost less at the dealership by checking which one is more expensive before and after 2.1 hours. Let's graph the lines and compare.
As we see in the graph, the blue line is below the red line before the point of intersection, which means the dealership costs are less than the costs at the local mechanic before 2.1 hours. Conversely, after 2.1 hours the local mechanic costs less.
Alternative Solution
Notice we could have also figured this out using logic. The amount the dealership charges for the parts is far less than the costs at the local mechanic. $24_(Dealership) < $45_(Local Mechanic)
Therefore, it is cheaper to use the dealership until the point that the hourly wage makes up for the cheaper parts.
