Problem 122 - College Algebra

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for \(\$ 50\) per day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

Step-by-Step Solution

Answer

One must drive more than 100 miles in a day for Basic Rental to become cheaper than Continental's.

Step 1: Set Up the Inequalities

We are given two rental companies with different pricing schemes. For Basic Rental, the cost can be expressed as \(C_b = 50 + 0.20m\), where \(m\) represents the mileage. Similarly, for Continental, the cost can be expressed as \(C_c = 20 + 0.50m\). We have to find a mileage point at which Basic's cost becomes less than Continental's, which is represented by \(C_b < C_c\).

Step 2: Solving the Inequality

Plugging the cost equations into the inequality, our inequality becomes: \(50 + 0.20m < 20 + 0.50m\). Firstly, we should subtract \(0.20m\) from both sides to isolate the remaining \(m\) term, giving us \(50 < 20 + 0.30m\). Then we subtract \(20\) from both sides, yielding \(30 < 0.30m\). Finally, dividing both sides by \(0.30\) leaves us with \(m > 100\).

Step 3: Interpret the Result

This result means that Basic Rental becomes a better deal than Continental when the mileage driven in a day exceeds 100 miles.