Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Two children own two-way radios that have a maximum range of 2 miles. One leaves a certain point at 1:00 P.M., walking due north at a rate of \(4 \mathrm{mi} / \mathrm{hr}\). The other leaves the same point at 1:15 P.M., traveling due south at \(6 \mathrm{mi} / \mathrm{hr}\). When will they be unable to communicate with one another?

Step-by-Step Solution

Answer

They will be unable to communicate at 1:21 PM.

Step 1: Understanding the Problem

We have two children moving in opposite directions from the same starting point. The first child walks north at a speed of 4 mph and starts at 1:00 PM. The second child walks south at a speed of 6 mph and starts at 1:15 PM. They are initially within a 2-mile communication range.

Step 2: Determine Each Child's Position Over Time

Let the position of the northbound child at time t hours after 1:00 PM be represented by the equation: \(x = 4t\). This child starts at 1:00 PM.Let the position of the southbound child t hours after 1:00 PM be represented by the equation: \(y = 6(t - 0.25)\) because this child starts 15 minutes (0.25 hours) later, at 1:15 PM.

Step 3: Establish the Distance Formula

The distance between the two children can be found using the formula for distance in a straight line since they move in opposite directions: \(D = x + y\). Therefore, \(D = 4t + 6(t - 0.25)\).

Step 4: Solve for When the Distance Equals the Maximum Range

We set \(D = 2\) (the maximum communication distance):\[4t + 6(t - 0.25) = 2\]Simplify and solve for t:\[4t + 6t - 1.5 = 2\]\[10t = 3.5\]\[t = 0.35\] hours after 1:00 PM.

Step 5: Convert Time from Decimal to Clock Time

To determine the exact time, convert 0.35 hours to minutes. 0.35 hours equals 21 minutes (since 0.35 x 60 minutes/hour = 21 minutes). Since this time is counted from 1:00 PM, the time when communication will fail is 1:21 PM.

Key Concepts

Two-Way Radios

Two-way radios are communication devices that allow people to talk to each other over a certain distance. Such devices are commonly used because of their convenience and efficiency.
They work by using radio waves to transmit voice messages.
In the context of this exercise, each child's two-way radio has a communication range of 2 miles. This means that if they move more than 2 miles apart, the radios will no longer be able to pick up each other's signals. Understanding this range is essential to solving the problem of when they will lose contact.

• The range of a two-way radio is typically determined by factors such as terrain, weather, and any potential obstructions.
• Two-way radios are particularly practical for short-range communication.
• This technology does not rely on cellular networks, making it ideal for remote areas.

By focusing on these key aspects, students can better understand why the 2-mile limitation is critical in determining when the children will lose their ability to communicate.

Linear Motion

Linear motion refers to movement along a straight path in one dimension. It is a straightforward concept that depicts objects moving in a straight line, such as the children in our exercise.
Both children start from the same location but move in opposite directions. One goes north, while the other goes south.
Understanding linear motion helps in calculating distances since both children travel along a single, straight trajectory.

• Linear motion is characterized by constant speed and direction.
• It simplifies calculations because there are no changes in directional vectors.
• Analyzing linear movement allows one to easily apply formulas for distance, speed, and time.

In this exercise, grasping the principle of linear motion is crucial because it simplifies finding the distance between the two children as they move further apart.

Distance Formula

The distance formula is a standard equation that helps calculate the space between two points moving in a straight line. For our problem, it is used to find when the distance between the two children exceeds their radios' range.
In the exercise, the equation is set up as:\[ D = 4t + 6(t - 0.25) \]Where \( x = 4t \) is the northbound child's movement and \( y = 6(t - 0.25) \) is the southbound child's motion.
By setting \( D = 2 \), we solve for \( t \), which tells us the time at which they will be 2 miles apart.

• The formula ensures accuracy in determining the distance using the known rates of travel.
• It allows students to understand how distance accumulates over time.
• This concept is widely applicable, enabling predictions of positions in various movement scenarios.

The distance formula is integral to solving not just this, but any distance-related problem involving linear paths and speed.

Rate of Change

Rate of change is a measure of how a variable changes over a certain time period. In the exercise, it refers to the speed at which each child is moving from the starting point.
It expresses how quickly the distance between the children and the starting point changes:

• The northbound child's rate is 4 mi/hr.
• The southbound child's rate is 6 mi/hr.

These rates of change are crucial in calculating the time needed for the children to exceed their radios' maximum range.
The concept of rate of change can be understood as a speed measure in mathematical terms:\[ ext{Rate of Change} = rac{ ext{Distance}}{ ext{Time}} \]Exploring rate of change in this scenario helps to emphasize how traveling in opposite directions affects their relative separation.

• It highlights the importance of speed in determining when communication is lost.
• Rate of change is essential for planning efficient travel routes.
• Understanding this concept supports other mathematical areas, including slopes in graphing.

By mastering rate of change, students can better predict dynamic systems and changes in various real-world situations.