Problem 38 - Algebra 1: Concepts and Skills

Question

Use the following information. You are climbing a 300 foot cliff. By 1: 00 P.M. you have climbed 110 feet up the cliff. By 3: 00 P.M. you have reached a height of 220 feet. Find the slope of the line that passes through the points \((1,110)\) and \((3,220)\) What does it represent?

Step-by-Step Solution

Answer

The slope of the line that passes through the points (1,110) and (3,220) is 55. This represents the rate of climbing, which is 55 feet per hour.

Step 1: Identify the given points

The given points are (1,110) and (3,220). Here, 1 and 3 represent the hours since 12:00 P.M., and 110 and 220 represent the height climbed at that time in feet.

Step 2: Apply the slope formula

Using the formula for the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\), calculate the slope, where (x1, y1) is (1, 110) and (x2, y2) is (3, 220). Therefore, \(m = \frac{220 - 110}{3 - 1} = \frac{110}{2} = 55\).

Step 3: Interpret the slope value

The slope of the line is 55, which means for each additional hour (increase in x), the height climbed (y) increases by 55 feet. Thus, the slope represents the rate of climbing, i.e., 55 feet per hour.

Key Concepts

Rate of Change

The rate of change is a fundamental concept in mathematics and real-world applications alike. It tells us how much one quantity changes in relation to another.
In our climbing example, we want to understand the rate at which you climb the cliff over time.When you have information about two different moments in time and how a particular measurement (like height) changes, you can calculate the rate of that change.
This is done using the slope formula, which in mathematical terms is \[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• In the climbing scenario, the rate of change in height, or the slope, is found by calculating how many feet you climb per hour.
• This is determined by looking at the difference in height (110 feet climbed) over a period of 2 hours.

This gives us a rate of 55 feet per hour, telling us that every hour, you climb 55 more feet than the previous hour.

Linear Equations

Linear equations form the basis of straight-line graphs. The equation of a line can be written generally as \[y = mx + b\]where \(m\) is the slope—the rate of change—and \(b\) is the y-intercept—the point where the line crosses the y-axis.
In our example, we deal with a linear relationship between time and height.

• The goal is to find a simple equation that represents this straight-line relationship.
• In our case, the 'x' values are the times in hours, and the 'y' values are the heights climbed.

The derived slope from our example, 55, can be used in our linear equation to predict how much will be climbed at any given time within the observed range.This type of equation allows us to easily understand and predict behavior based on the rate of change.

Coordinate Points

Coordinate points are pairs of numbers that give information about positions on a plane, specifically on the Cartesian coordinate system.
In the task at hand, these points represented time and height.

• Each point is written as \((x, y)\), where \(x\) is the independent variable and \(y\) is the dependent variable.
• In our climbing exercise, \((1, 110)\) and \((3, 220)\) tell us what height, \(y\), corresponds to which hour, \(x\).

Understanding these points is crucial as they are used in calculating the slope and interpreting linear movement. Applying these coordinates properly helps visualize the relationship between variables, making the problem-solving process more manageable.