Problem 132 - College Algebra

Question

Parts for an automobile repair cost \(\$ 175 .\) The mechanic charges \(\$ 34\) per hour. If you receive an estimate for at least \(\$ 226\) and at most \(\$ 294\) for fixing the car, what is the time interval that the mechanic will be working on the job?

Step-by-Step Solution

Answer

The time interval that the mechanic will be working on the job is approximately between 1.5 and 3.5 hours.

Step 1: Understand the Problem

The cost of car parts is fixed at $175. The mechanic charges $34 per hour, so the cost of labor is $34t, where t is the number of hours the mechanic works. The total cost of the repair is the sum of the cost of parts and the cost of labor, hence it will be $175 + $34t. This total cost must be at least $226 but no more than $294.

Step 2: Set Up Inequalities

Set up two inequalities to represent the cost limits. The first inequality for the lower limit would be: $175 + $34t ≥ $226. The second inequality for the upper limit would be: $175 + $34t ≤ $294.

Step 3: Solve the Lower Limit Inequality

To solve the first inequality $175 + $34t ≥ $226, subtract $175 from both sides to isolate $34t, which gives $34t ≥ $51. Next, divide each side by $34 to solve for t, which gives \( t \geq \frac{51}{34} \approx 1.5 \) hours.

Step 4: Solve the Upper Limit Inequality

To solve the second inequality $175 + $34t ≤ $294, subtract $175 from both sides to get $34t ≤ $119. Then, divide each side by $34 to solve for t, which gives \( t \leq \frac{119}{34} \approx 3.5 \) hours.