Problem 32 - Algebra 1

Question

Write a linear equation to model the situation. Use unit analysis to check your model. You have walked 5 miles on a hiking trail. You continue to walk at the rate of 2 miles per hour for 6 hours.

Step-by-Step Solution

Answer

The linear equation that models the hiking situation is \(y = 2x + 5\), and for 6 hours of additional walking, the total distance walked would be 17 miles.

Step 1: Identify the quantities

Identify the quantities and their values: Initial distance traveled = 5 miles. Walking speed = 2 miles/hour. Time spent walking = 6 hours.

Step 2: Write the linear equation

Establish a linear equation that includes both the initial distance and the distance traveled during the additional walking time. The equation takes the form of \(y = mx + c\), where \(y\) is the total distance traveled, \(m\) is the speed (rate of walking), \(x\) is the time spent walking, and \(c\) is the initial distance. From the values given, the linear equation becomes: \(y = 2x + 5\).

Step 3: Unit analysis

Check the model using unit analysis. The units in the equation should be consistent on both sides. In this case, since 'miles' = 'miles/hour' * 'hours' + 'miles', the units on both sides of the equation are indeed consistent.

Step 4: Output the total distance traveled

Calculate the total distance traveled after the additional 6 hours: \(y = 2*6 + 5\).

Key Concepts

Unit Analysis

When working with linear equations, unit analysis is an essential tool to verify accuracy. This technique involves examining and ensuring consistency of units across an equation. It's like a double-check to avoid mistakes.

In the given problem, the equation models a situation where distance is calculated using rate and time. The form of the equation is: - Total distance traveled (\( y \)) on the left side, expressed in miles.- On the right side, you have terms like initial distance (5 miles) and your walking rate "miles per hour" (\( 2 \)) multiplied by time "hours" (\( x \)). Unit analysis checks if multiplying the walking rate by time results in a distance with consistent units on both sides.

Breaking it down:- Rate (\( 2 \) miles per hour) times time (\( x \), hours) gives a product in miles.- Adding any remaining miles results in the same unit (miles) as the left side of the equation.This simple method ensures that your math checks out before you go on to calculate answers.

Distance Calculation

Understanding how distance is calculated in this scenario involves the creation of an equation that adds up all distances. Initially, you walked 5 miles, and then you continued hiking at a steady rate for more time.

Here's how the calculation works:- You begin with a known distance, 5 miles.- For the additional walking, use the rate of walking (2 miles per hour) and multiply by the time (6 hours).- This product gives you the additional distance traveled, which needs to be added to the starting miles.

The linear equation reflecting this situation is:\(y = 2x + 5\)- Where \( y \) is the total distance traveled.The calculation breaks down to:\[ y =2 imes 6 + 5\]You obtain 12 miles from the walking part, adding it to your initial 5 miles gives a total of 17 miles.

Rate and Time Relationship

The relationship between rate and time forms the central part of many problems around linear equations. Especially in scenarios involving motion.

The rate, often given in units like miles per hour, tells you how fast or slow an activity is progressing. Time quantifies how long this activity lasts.
Let’s break it down:- **Understanding Rate**: This is the 'speed' part of your linear equation. For instance, hiking at 2 miles/hour means covering 2 miles in one hour.- **Understanding Time**: This is the duration aspect. Continuing hiking for 6 hours at that speed is going to show you how far you went during the additional walk.
When combined, rate and time in this equation: \( y = mx + c \)where: - \( m \) stands for rate,- \( x \) represents time, Illustrate how much extra distance contributions are based on the speed and duration of an activity.