Problem 24 - Algebra and Trigonometry with Analytic Geometry

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 runs 6.92 miles in total.

Step 1: Determine Total Jogging Time

The woman starts jogging at 3:00 P.M. and returns at 3:45 P.M. Total jogging time is the difference: 45 minutes. So, she jogs for 45 minutes in total.

Step 2: Set Up Equations for Distances

Let the distance she jogs north be \(d_1\) and the distance south be \(d_2\). She reverses direction and ends at her start point, so \(d_1 = d_2\). We need to find \(d_1 + d_2\).

Step 3: Define Duration of North and South Segments

At 6-minute-mile pace, the time she jogs north is \(t_1\), and at 7-minute-mile pace, the time she jogs south is \(t_2\), with \(t_1 + t_2 = 45\) minutes.

Step 4: Express Distances in Terms of Time

For northbound, \(d_1 = \frac{t_1}{6}\) since she runs 1 mile every 6 minutes. For southbound, \(d_2 = \frac{t_2}{7}\) since she runs 1 mile every 7 minutes.

Step 5: Solve for Time Ratios

Since \(d_1 = d_2\), then \(\frac{t_1}{6} = \frac{t_2}{7}\). Cross-multiplying gives \(7t_1 = 6t_2\), leading to \(t_2 = \frac{7}{6}t_1\). Substitute back into \(t_1 + t_2 = 45\): \(t_1 + \frac{7}{6}t_1 = 45\).

Step 6: Calculate Individual Times

Simplifying, \(\frac{13}{6}t_1 = 45\) leads to \(t_1 = \frac{6}{13} \times 45 = 20.77\) minutes. Thus, \(t_2 = 45 - 20.77 = 24.23\) minutes.

Step 7: Calculate Total Distance Run

Now calculate \(d_1\) and \(d_2\).\(d_1 = \frac{t_1}{6} \approx \frac{20.77}{6} = 3.46\) miles. Similarly, \(d_2 = \frac{t_2}{7} \approx \frac{24.23}{7} = 3.46\) miles. Thus, total distance = \(d_1 + d_2 = 6.92\) miles.

Key Concepts

Distance and Speed

Understanding distance and speed is crucial for solving many algebraic problems, particularly those involving motion. In this jogging exercise, the woman's pace is pivotal. Her speed northbound is 1 mile every 6 minutes, and southbound it is 1 mile every 7 minutes.
In general:

• Speed indicates how fast an object is moving, usually stated as miles per hour or another unit of distance per unit of time.
• Distance is how far an object travels, calculated by multiplying speed by the time traveled.

For example, if you run at a speed of 6-minute miles for 18 minutes, distance is calculated as \( \frac{18}{6} = 3 \) miles.
This concept helps us break complex problems into manageable pieces by separately analyzing motion in different directions.

Equations and Inequalities

Equations and inequalities are mathematical expressions used to find unknown values. In the jogger's problem, equations help determine how much time she spends jogging north and south.
Here are the steps taken:

• Assign variables such as \( t_1 \) for the jog north and \( t_2 \) for the jog south.
• Using that she returns to the starting point, we know distances traveled north and south are equal: \( d_1 = d_2 \).
• Express the distances as equations: \( d_1 = \frac{t_1}{6} \) and \( d_2 = \frac{t_2}{7} \).

Inequalities are not directly used here since the problem requires exact values, but they can help identify relationships when approximations or ranges are involved.
For instance, one might use inequalities to determine if one leg of the journey was longer under different constraints.

Time Management in Computations

Managing time calculations is a key part of solving algebraic problems involving motion. In the problem, the total time available is 45 minutes, split between northbound and southbound jogging.
Let's see how to manage this:

• Understand the relationship between time and distance: more time at a slower pace results in a longer path covered.
• Calculate time efficiently by substituting one expression into another: here, \( t_2 = \frac{7}{6}t_1 \) simplifies the solution.

It's crucial to ensure the total time adds up correctly: \( t_1 + t_2 = 45 \). By setting up and solving equations, you allocate time accurately to each segment of the jog.
This approach assists in visualizing how time is distributed in real-world applications, enabling one to solve problems faster and with less error.