Problem 132 - College Algebra Essentials

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. 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 mechanic will be working on the job for a timeframe between approximately 1.5 hours and 3.5 hours.

Step 1: Identifying the Knowns

The cost of the parts is fixed at \$175. The hourly rate of the mechanic's labor is \$34 per hour. The total estimated cost for the job lies within the range \(\$226 \leq Total \leq $294\). The total cost is a sum of the cost of the parts and labor, which can be expressed as \(Total = $175 + $34*hours\).

Step 2: Forming the Inequalities

Using the equation in Step 1, the minimum and maximum costs can be represented as two separate inequalities. For the minimum cost, we have \(226 \leq 175 + 34 * hours\). And for the maximum cost, we have \(175 + 34 * hours \leq 294\).

Step 3: Solving the Inequalities

Each inequality is solved for 'hours' to find its value. Begin by subtracting 175 from both sides. For the minimum cost, the equation is: \(51 \leq 34 * hours\). Dividing each side by 34, we get: \(hours \geq \frac{51}{34} \approx 1.5\) hours. For the maximum cost, the inequality is: \(119 \geq 34 * hours\). Dividing each side by 34, we get: \(hours \leq \frac{119}{34} \approx 3.5\) hours.