Problem 74 - Precalculus: Mathematics for Calculus

Question

Aerobic Exercise \(\quad\) A woman keeps fit by bicycling and running every day. On Monday she spends \(\frac{1}{2} \mathrm{h}\) at each activity, covering a total of \(12 \frac{1}{2} \mathrm{mi}\). On Tuesday she runs for 12 min and cycles for 45 min, covering a total of 16 mi. Assuming that her running and cycling speeds don't change from day to day, find these speeds.

Step-by-Step Solution

Answer

Running speed is 5 mph, and cycling speed is 20 mph.

Step 1: Interpret the Problem

The problem gives us two activities: running and cycling. We know the times and total distances for Monday and Tuesday. Our goal is to find the speeds for both activities.

Step 2: Establish Variables

Let \( r \) represent the running speed in miles per hour (mph), and \( c \) represent the cycling speed in mph.

Step 3: Convert Time to Hours

On Monday, each activity takes 0.5 hours. On Tuesday, the times must be converted from minutes to hours: 12 minutes is 0.2 hours, and 45 minutes is 0.75 hours.

Step 4: Write Equations for Monday

The total distance equation for Monday is: \( 0.5r + 0.5c = 12.5 \).

Step 5: Write Equations for Tuesday

The total distance equation for Tuesday is: \( 0.2r + 0.75c = 16 \).

Step 6: Solve for One Variable

From the Monday equation, solve for \( r \): \( r = 25 - c \).

Step 7: Substitute into Tuesday Equation

Substitute \( r = 25 - c \) into the Tuesday equation: \( 0.2(25 - c) + 0.75c = 16 \).

Step 8: Simplify the Tuesday Equation

Distribute and simplify: \( 5 - 0.2c + 0.75c = 16 \), resulting in \( 0.55c = 11 \).

Step 9: Solve for Cycling Speed

Divide both sides by 0.55 to find \( c \): \( c = 20 \).

Step 10: Solve for Running Speed

Substitute \( c = 20 \) into \( r = 25 - c \): \( r = 25 - 20 = 5 \).

Step 11: Verify the Solution

Check the values by substituting back into both original equations to ensure they hold true.

Key Concepts

Speed Calculation

Understanding how to calculate speed is essential for solving distance problems effectively. Speed can be thought of as the rate at which an object covers a distance. It is usually calculated in terms of miles per hour, kilometers per hour, or meters per second.
To calculate speed, use the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
In the exercise above, we need to find the running and cycling speeds of the woman. This involves using the relationship between distance, time, and speed. The distances for each activity are provided, as well as the time spent on these activities each day. With these, we can determine the speed of each activity by rearranging the speed equation to solve for the unknown variables of speed.
Always remember: Keeping your units consistent (e.g., hours and miles) is crucial for accuracy when calculating speed.

System of Equations

A system of equations is a set of two or more equations with the same variables. Solving these equations simultaneously helps us find an unknown value.
In this exercise, we have two activities (running and cycling) that occur over two days, leading us to form a system of equations based on the given data. Our goal is to determine the running and cycling speeds.
For Monday, the equation using the given total distance and time for both activities is:
\[ 0.5r + 0.5c = 12.5 \]
For Tuesday, using the times converted to hours and the total distance is:
\[ 0.2r + 0.75c = 16 \]
These equations create a system, which we solve by substitution or elimination methods to find the values of the speeds. By carefully isolating one variable and substituting back, we find the solution step-by-step. This systematic approach helps ensure we accurately determine both values.

Unit Conversion

Unit conversion is a key part of solving problems involving distance, rate, and time. It's important because we often measure time in different units such as minutes and hours, or distance in miles and kilometers.
In this particular problem, the woman bikes and runs for different durations. To use these in the equations, we must convert the time spent from minutes to hours. The conversion factor is:
\[ 1 \text{ hour} = 60 \text{ minutes} \]
For example, 12 minutes becomes \( 0.2 \) hours and 45 minutes becomes \( 0.75 \) hours. By converting these units, we ensure that all our calculations are in the same unit, allowing us to find an accurate solution.
Proper unit conversion not only aids in achieving correct answers but also in maintaining consistency throughout equations and computations.