Problem 57 - Algebra 1: Concepts and Skills

Question

Use the following information. Snow fell for 9 hours at a rate of \(\frac{1}{2}\) inch per hour. Before the snowstorm began, there were already 6 inches of snow on the ground. The equation \(y=\frac{1}{2} x+6\) models the depth y (in inches) of snow on the ground after x hours. Explain what the slope and y-intercept represent in the snowstorm model.

Step-by-Step Solution

Answer

In the snowstorm model, the slope of \(\frac{1}{2}\) represents the rate at which snow is falling, at \(\frac{1}{2}\) inch per hour. The y-intercept of 6 represents the depth of snow that was already on the ground before the snowstorm began, at 6 inches.

Step 1: Interpret the slope

The given equation has the form of \(y=mx+c\), where m is the slope and c is the y-intercept. Here, the slope is \(\frac{1}{2}\). In the context of this snowstorm model, the slope represents the rate at which the snow is falling, measured in inches per hour. So, a slope of \(\frac{1}{2}\) means that the snow is falling at a rate of \(\frac{1}{2}\) inch per hour.

Step 2: Interpret the y-intercept

The y-intercept of the equation is 6. The y-intercept represents the value of y when x is zero, that is, before the snowstorm began. So, in the context of this model, the y-intercept (6 inches) represents the depth of the snow that was already on the ground before the snowstorm began.

Key Concepts

Slope

When discussing linear equations, the concept of a slope is essential to understanding how variables relate to each other. In the equation of a line, which is often written as \(y=mx+c\), the slope is represented by \(m\). This coefficient reflects the rate of change or how much \(y\) changes when \(x\) increases by one unit.

In the context of the snowstorm exercise, the slope \(\frac{1}{2}\) indicates that snow is accumulating at a rate of \(\frac{1}{2}\) inch per hour. Essentially, for each hour that passes, the snow depth increases by half an inch. This information helps to model the ongoing rate of snowfall over time. Knowing the slope helps predict how much snow will accumulate after a certain number of hours based on this steady rate.

Y-Intercept

The y-intercept is another vital component of a linear equation, expressed as \(c\) in the formula \(y=mx+c\). This value describes where the line crosses the y-axis. In practical terms, it's the value of \(y\) when \(x\) is zero.

Relating this to the snowstorm problem, the y-intercept is 6, which tells us that there were already 6 inches of snow on the ground before the snowstorm commenced. When modeling real-world situations, understanding the y-intercept provides insight into initial conditions or starting values, offering a complete picture of how the model behaves over time.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena through mathematical equations and expressions. This method allows us to predict and interpret trends, making it especially useful in scenarios like weather analysis or economic forecasting.

In the case of the provided snowstorm exercise, the linear equation \(y=\frac{1}{2}x+6\) models how snow accumulates over time. Here, two main components, the slope and y-intercept, are used to accurately depict the relationship between time and the depth of the snow.

• The slope \(\frac{1}{2}\) illustrates the snowfall rate, providing essential data for expected snow depth after certain hours.
• The y-intercept 6 gives us the baseline snow level, a critical initial condition that affects total snow accumulation.

Understanding these parts of a mathematical model lets us describe and predict outcomes with precision, making such models indispensable in both theoretical studies and practical applications.