Problem 9 - Algebra 1: Concepts and Skills

Question

In Exercises \(7-11\), a mountain climber is scaling a 400 -foot cliff. The climber starts at the bottom at \(t=0\) and climbs at a constant rate of 124 feet per hour. Use the slope and \(y\) -intercept to write the linear model for the distance \(y\) (in feet) that the climber climbs in terms of time \(t\) (in hours). Use slope intercept form.

Step-by-Step Solution

Answer

The linear model for the distance the climber climbs in terms of time is \(y = 124t\).

Step 1: Identify the slope and the y-intercept

The climber climbs at a constant rate of 124 feet per hour. This pace of climbing is equivalent to the slope (m) of the linear equation. The climber starts at the bottom of the cliff which means at t=0, y=0. This is the y-intercept (c). So, m = 124 and c = 0.

Step 2: Plug values into slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + c\). If we substitute the values of m and c that we identified in the previous step, we get \(y = 124t + 0\).

Step 3: Simplify the equation

The equation \[y = 124t + 0\] can be simplified to \(y = 124t\), because adding 0 does not change the value of 124t.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most popular ways to express linear equations. If you're dealing with linear equations, the slope-intercept form will always come in handy. This form is given by the equation \( y = mx + c \), where:

• \( y \) is the dependent variable (usually representing the output value).
• \( m \) is the slope of the line.
• \( x \) is the independent variable (often representing time or input).
• \( c \) is the y-intercept.

When using this form, the main goal is to identify the slope and y-intercept from the problem to plug into the equation. In the context of the mountain climber, you're essentially creating a model for how the climber's distance changes over time. By identifying both the rate of climbing (the slope) and the starting point (the y-intercept), you can craft an equation that mirrors this relationship.

Rate of Change

The rate of change is a concept that you often hear about in math and science, particularly when working with graphs and functions. In linear equations like the one used for the mountain climber, the rate of change is referred to as the “slope.” A constant rate of change means that the variable changes by the same amount over each equal interval. Here:

• A slope of 124 feet per hour means that for every hour that passes, the climber ascends an additional 124 feet.
• This constant, unchanging rate highlights how the position of the climber increases uniformly with time.

Unlike more complex functions that might curve or dip, linear functions with constant rates of change are straight lines when graphed. In this exercise, understanding that the climber's rate remains constant allows you to accurately model and predict how much they’ll climb over any given period.

Y-Intercept

The y-intercept is a critical point on a graph that reveals where a line crosses the y-axis, corresponding to \( x = 0 \). It's an essential part of understanding linear equations, as it gives the starting point for the function in relation to the vertical axis. In our mountain climbing example:

• At \( t = 0 \) hours, the climber hasn't yet ascended any part of the cliff. That's 0 feet, hence the y-intercept is 0.
• This ensures the line accurately starts at the origin point, reflecting real-life parameters where time begins and the climb starts!

Understanding the y-intercept is crucial because it gives your linear equation a specific starting point. It helps to visualize and sketch out the graph of the linear model, making the connection between the theoretical math problem and the real-world scenario more apparent.