Problem 47 - Intermediate Algebra

Question

At 2 P.M., two military convoys leave Eagle River, Wisconsin, one headed north and one headed south. The convoy headed north averages \(50 \mathrm{mph}\), and the convoy headed south averages 40 mph. They will lose radio contact when the distance between them is more than 35 miles. When will this occur?

Step-by-Step Solution

Answer

The convoys will lose radio contact at approximately 2:23 P.M.

Step 1: Define the Variables

First, let's define the variables needed to solve this problem. Let \( t \) represent the time in hours after 2 P.M. We want to find the time \( t \) when the convoys are more than 35 miles apart. The northbound convoy travels at 50 mph and the southbound convoy travels at 40 mph.

Step 2: Write Distance Formulas

The distance each convoy travels is given by the formula: \( \text{Distance} = \text{Rate} \times \text{Time} \). Therefore, the distance traveled by the northbound convoy is \( 50t \) miles and the distance traveled by the southbound convoy is \( 40t \) miles.

Step 3: Setup the Distance Equation

The total distance between the two convoys after \( t \) hours is the sum of their distances: \( 50t + 40t = 90t \). They lose radio contact when their distance is more than 35 miles. So, we set up the inequality: \( 90t > 35 \).

Step 4: Solve the Inequality

To find \( t \), solve the inequality: \( 90t > 35 \). Divide both sides by 90 to isolate \( t \):\[ t > \frac{35}{90} \] This simplifies to \( t > \frac{7}{18} \) hours.

Step 5: Convert Time to Minutes

Now, we convert \( \frac{7}{18} \) hours to minutes. Since there are 60 minutes in an hour, calculate:\[ t > \frac{7}{18} \times 60 \approx 23.33 \] minutes. Thus, the convoys will lose radio contact at approximately 2:23 P.M.

Key Concepts

Military Convoys

Military convoys are groups of vehicles, often large and carrying essential supplies or personnel, that travel together for mutual support and protection. When these convoys are on a mission, maintaining constant communication is crucial for coordination and safety. In our example, two military convoys depart from the same location, heading in opposite directions.
They remain within radio contact until they are separated by a specific distance threshold, in this case, 35 miles. This particular problem is a scenario where mathematics helps us predict when the radio contact might be lost due to the increasing distance between convoys traveling at known speeds.

Inequality Solving

Inequality solving involves finding the range of values that satisfy a given inequality, rather than an exact solution which is often the case with equations. In our context, we set up the inequality based on the total distance between two convoys.
We express this with:

• The northbound convoy travels at 50 mph and the southbound at 40 mph.
• After time \( t \), the total separation is \( 90t \) miles.
• We set this greater than 35 miles: \ (90t > 35 ) \.

To solve the inequality, divide both sides by 90 to isolate \( t \):\[t > \frac{35}{90}\]Simplifying gives \( t > \frac{7}{18} \) hours, indicating when the convoys will exceed the specified distance.

Rate and Time

Rate and time are fundamental concepts in distance problems, and they relate directly to the calculation of how far something travels over a period. The relationship is given by the formula:

• \( \text{Distance} = \text{Rate} \times \text{Time} \)

In this problem:

• The northbound convoy travels at 50 mph.
• The southbound convoy travels at 40 mph.

The total time in hours determines the distance each convoy covers. For instance, if \( t = 1 \), the northbound convoy travels 50 miles; whereas if \( t = 2 \), it travels 100 miles.This linear relationship helps in setting up and solving the equation concerning how much the convoys are apart after a specified time.

Radio Contact Distance

Radio contact distance refers to the maximum distance over which reliable radio communication can be maintained between two points. Beyond this distance, radio signals will weaken and eventually fail, causing the loss of communication. This maximum range can vary based on terrain, weather, and equipment used. In the exercise, the convoys lose contact when separated by more than 35 miles.
Understanding radio contact distance is essential for operations requiring uninterrupted communication. It helps in planning the routes and the logistics of convoys to ensure that all parts of a mission remain informed and coordinated until the point where communication is predicted to be compromised. This prediction is based on mathematical calculations, as demonstrated in the exercise.