Problem 58 - Algebra 1

Question

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 \(y\) of snow on the ground after \(x\) hours. Graph the amount of snow on the ground during the storm.

Step-by-Step Solution

Answer

The graph of the equation \(y=\frac{1}{2} x+6\) over 9 hours shows a steady increase in snow depth from 6 inches to 10.5 inches. The slope of \(\frac{1}{2}\) indicates the snowfall rate per hour, and the y-intercept at 6 indicates the initial depth of the snow.

Step 1: Identify the slope and y-intercept

Looking at the equation \(y=\frac{1}{2}x+6\), we can identify the slope and the y-intercept. The number in front of \(x\), \(\frac{1}{2}\), is the slope, which tells us that the snow is falling at a rate of \(\frac{1}{2}\) inch per hour. The y-intercept is 6, which represents the initial amount of snow on the ground before the snowfall started. This is where the line will cross the y-axis on the graph.

Step 2: Draw the y-intercept

First, draw a pair of perpendicular lines to represent the x-axis (time in hours) and the y-axis (depth in inches). Next, locate 6 on the y-axis and make a point. This represents the depth of the snow before the snowfall started.

Step 3: Draw the slope

From the point at 6 on the y-axis, which corresponds to the y-intercept, move up \(\frac{1}{2}\) inch (the slope) for each step forward of 1 hour along the x-axis. Make another point for each hour up to 9 hours. This is because the snow fell for 9 hours.

Step 4: Draw the line

Draw a straight line that passes through all the points plotted on the graph. This line represents the depth of snow over the 9 hour period.

Step 5: Interpret the graph

The graph shows how the depth of the snow increased at a steady rate of \(\frac{1}{2}\) inch per hour over the 9 hour period. At any point along the line, the x-coordinate represents the number of hours into the snowfall, and the y-coordinate represents the total depth of the snow at that time.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear function is crucial for graphing and analyzing linear relationships. In essence, this is the equation of a line written as
\( y = mx + b \).
The 'm' stands for slope, which is the rate at which y increases as x increases. The 'b' in the equation represents the y-intercept, which is the point where the line crosses the y-axis. In our snowfall example,
\( y = \frac{1}{2}x + 6 \),
the slope is \( \frac{1}{2} \) indicating that for every hour passing, there's half an inch of snowfall. The y-intercept is 6, telling us that before the snow began to fall, there were already 6 inches of snow on the ground. These two values give us all the information we need to graph the equation and understand how the depth of snow changes over time.

Graphing Linear Equations

Graphing linear equations like
\( y = \frac{1}{2}x + 6 \)
allows students to visualize the problem and see the direct relationship between variables. To graph a linear equation, you start by plotting the y-intercept on the y-axis. In this case, you would plot a point at 6 on the y-axis. Next, you use the slope, or rate of change, to find other points that lie on the line. With a slope of \( \frac{1}{2} \), you move up half a unit for every one unit you move to the right. By repeatedly applying the slope, you can plot several points that fulfill the equation — these points should form a straight line. After plotting these points up to the given time of 9 hours, draw a line through all the points to visualize how the snow depth increases every hour.
Graphical representation not only makes the problem easier to understand but also allows for visual interpretation of data, making it a valuable skill in various fields of study and professional work.

Rate of Change

The rate of change in a linear equation is synonymous with the slope and it describes how one variable changes in relation to another. In the context of our exercise, the slope
\( \frac{1}{2} \) inches per hour
is the rate of change of snow depth per hour. It indicates a consistent, uniform increase in snow depth for each hour of time that passes. This steady rate tells you that no matter which hour you look at, you can expect an additional half inch of snow from the previous hour. Understanding the rate of change is fundamental in predicting outcomes and in making informed assumptions based on the linear model, such as estimating the depth of snow after any given time during the snowstorm.