Problem 82 - Intermediate Algebra

Question

The distance that a truck can travel in 8 hours, at a constant rate of \(r \mathrm{mph},\) is given by \(8 r .\) A trucker wants to travel at least 350 miles, and company regulations don't allow him to exceed 450 miles in one 8 -hour shift. a. Complete the double inequality that describes the mileage range of the truck. \(\leq 8 r \leq\) b. Solve the double inequality to find the range of the average rate (speed) of the truck for the 8 -hour trip.

Step-by-Step Solution

Answer

a. \( 350 \leq 8r \leq 450 \); b. \( 43.75 \leq r \leq 56.25 \).

Step 1: Set Up the Inequality

We need to complete the inequality to represent the mileage range for the truck. Since the trucker wants to travel at least 350 miles, the inequality begins with 350. Therefore, the inequality is: \( 350 \leq 8r \leq 450 \).

Step 2: Divide Each Part by 8

To isolate \( r \), divide the entire inequality by 8. This involves dividing each part: \( \frac{350}{8} \leq \frac{8r}{8} \leq \frac{450}{8} \).

Step 3: Simplify the Inequality

Calculate the results of the divisions: \( 43.75 \leq r \leq 56.25 \). This gives us the range of the average speed \( r \).

Key Concepts

Distance-Rate-Time Problems

Understanding how distance, rate, and time relate is crucial for many real-life scenarios. The fundamental formula that ties these elements together is \( \,\text{Distance} = \text{Rate} \, \times \, \text{Time} \,\). This formula is incredibly versatile and is used to calculate any of the three variables if the other two are known.In our problem, the truck travels a certain distance within a specific time frame. Here, the time is fixed at 8 hours, and the rate is given as \( r \, \text{mph} \). To find the total distance, we multiply the rate by time, resulting in \( 8r \, \text{miles} \). Let's break it down with key points:

• Distance: How far an object moves, measured at \( 8r \, \text{miles} \) in this example.
• Rate: The speed at which the object travels, given as \( r \, \text{mph} \).
• Time: The duration of travel, 8 hours here.

These concepts help in setting constraints for practical problems like our truck scenario where the number of miles driven is based on a speed range.

Double Inequalities

Double inequalities tackle problems where a value must stay within certain bounds. They're a compact way to express two inequalities simultaneously. In our scenario, the truck's mileage in 8 hours is bounded between a minimum and maximum value.Here's a closer look at the example:

• The Lower Bound: The truck must travel at least 350 miles, forming the part of the inequality \( 350 \leq 8r \).
• The Upper Bound: Company rules cap the miles at 450, creating the inequality \( 8r \leq 450 \).
• Combined as: \( 350 \leq 8r \leq 450 \).

Double inequalities ensure that solutions respect both conditions simultaneously.

Solve Inequalities

Solving inequalities involves finding the values that make the inequality a true statement. Our main goal is to isolate the variable of interest—in this case, \( r \) for the truck's rate.Starting with the inequality \( 350 \leq 8r \leq 450 \):

• To isolate \( r \), we perform the same operation on all parts of the inequality. Here, dividing by 8 simplifies the expression.
• We calculate: \( \frac{350}{8} \leq r \leq \frac{450}{8} \).
• Simplifying gives \( 43.75 \leq r \leq 56.25 \).

This tells us that the truck can travel at any average speed between 43.75 mph and 56.25 mph to meet both the minimum and maximum distance requirements in the 8-hour period. Solving inequalities requires balancing each side by performing the same operations, ensuring all steps lead to a true statement.