Problem 70 - Precalculus: Functions and Graphs

Question

A woman begins jogging at 3: 00 P.M., running due north at a 6-minute-mile pace. Later, she reverses direction and runs due south at a 7 -minute-mile pace. If she returns to her starting point at 3: 45 P.M., find the total number of miles run.

Step-by-Step Solution

Answer

The woman jogged about 6.92 miles.

Step 1: Understanding the Problem

The woman jogs north at 6-minute-per-mile pace and then reverses to jog south at 7-minute-per-mile pace. She starts at 3:00 PM and returns by 3:45 PM. We need to find out how many miles she jogged in total.

Step 2: Determine the Total Time

The woman jogs from 3:00 PM to 3:45 PM. Calculate the total jogging time, which is 45 minutes.

Step 3: Define Variables

Let the distance north be \(x\) miles. Then the same distance \(x\) is covered going south. We need to calculate \(2x\), the total distance jogged.

Step 4: Calculate Time Going North

At a pace of 6 minutes per mile, the time to run \(x\) miles north: \(6x\) minutes.

Step 5: Calculate Time Going South

At a pace of 7 minutes per mile, the time to run \(x\) miles south: \(7x\) minutes.

Step 6: Set Up Equation

The total time for northward and southward jogging is given by: \(6x + 7x = 45\) minutes.

Step 7: Solve the Equation

Combine terms in the equation: \(13x = 45\). Solve for \(x\) to find \(x = \frac{45}{13}\).

Step 8: Calculate Total Distance

The total jogging distance is \(2x = 2 \times \frac{45}{13} = \frac{90}{13} \approx 6.92\) miles.

Key Concepts

Pace Calculation

When we talk about pace in running, we are essentially discussing the time it takes to cover a specific distance. The pace is expressed in minutes per mile or kilometer. This is a crucial concept for understanding how runners manage their speed and distance over time. In the original exercise, the woman runs consistently north with a pace of 6 minutes per mile and south at 7 minutes per mile.

To determine how fast she is going, we can easily understand the pace as the inverse of speed. This means that a lower number in a minute-per-mile pace signifies a faster speed. For instance:

• A 6-minute-per-mile pace means each mile is covered in 6 minutes.
• A 7-minute-per-mile pace means each mile is covered in 7 minutes.

By setting this pace, we can predict how long it will take to cover each section of her run. It's a simple, yet powerful way to link time with the distance in running scenarios.

Time and Distance Relationship

Time and distance are interconnected through pace. They always work together in any movement scenario. The relationship can be captured by the basic formula: Distance = Speed x Time (or in pace terms: Distance = (Time / Pace)).

In the woman's jogging case, we can break it down:

• She jogs north at 6 minutes per mile, determining the time required is essential to find out how far she moves in a straight line northward.
• The south-running pace is 7 minutes per mile, which affects her return time to the starting point.
• Total jog duration is from 3:00 PM to 3:45 PM, or 45 minutes.

Understanding this relationship allows us to calculate how each section (north and south) of her run fits into the total time frame by using equations and known variables.

Linear Equations in Problem Solving

Linear equations can be very handy in breaking down problems involving pace and distance. They provide a means of translating a verbal problem into a mathematical model and solve for unknowns through reasoning and calculations.

In this exercise, we use linear equations to find the total distance jogged. Here's how it breaks down:

• Assume the northward jog is for a distance of \(x\) miles. Then the return (south) jog is also \(x\) miles.
• The total time for north and south jogging is expressed as: \(6x + 7x = 45\) minutes (since north is at 6 min/mile and south at 7 min/mile).
• This translates to \(13x = 45\). Solving for \(x\), we find \(x = \frac{45}{13}\).
• The total distance is thus \(2x = 2 \times \frac{45}{13} = \frac{90}{13} \approx 6.92\) miles.

This example showcases how effectively setting up an equation allows for a precise and logical solution to what may seem a complex real-world problem.