Problem 59 - 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. Writing In the Apple County School District, school is canceled if there is a foot or more of snow on the ground. If this 9-hour storm occurred in Apple County, would school be canceled? Explain your reasoning.

Step-by-Step Solution

Answer

No, school would not be canceled in Apple County. The total snowfall of 10.5 inches after the 9-hour storm is less than a foot, so it does not meet the district's cancellation conditions.

Step 1: Understanding of Problem and Equation

The equation \(y=\frac{1}{2}x+6\) represents the total depth of the snow. Here, \(x\) represents the number of hours the snowfall lasts and \(\frac{1}{2}x\) is the amount of snow that falls each hour. The \(+6\) represents the snow that was already on the ground before the snowstorm started.

Step 2: Substituting Given Values

Now, by substituting the given value of \(x=9\) hours into the equation, we find the total snow depth. So, \( y = \frac{1}{2} * 9 + 6 \)

Step 3: Calculating Total Snow Depth

Carrying out the calculation, \( y = 4.5 + 6 = 10.5 \) inches.

Step 4: Comparing With the Cancellation Condition

Apple county cancels school if there is a foot or more of snow. A foot is equivalent to 12 inches. So, comparing 10.5 inches with 12 inches, the snowfall is less than a foot.

Step 5: Making Conclusion

Since the total snowfall, 10.5 inches, is less than 12 inches (a foot), according to the rules of the Apple County School District, school would not be canceled.

Key Concepts

Snowfall Calculation

When calculating snowfall during a storm, we need to consider both the rate of snow accumulation and any initial snow depth before the storm began. In this exercise, snow fell at a steady rate of \(\frac{1}{2}\) inch per hour. This means that for every hour of snowfall, half an inch is added to the already existing snow layer.

There were already 6 inches of snow on the ground before the storm started. The linear equation given is \(y = \frac{1}{2}x + 6\), where:

• \(y\) is the total depth of snow, in inches, at any given time.
• \(x\) is the number of hours that the snow has been falling.
• \(\frac{1}{2}x\) gives the new snowfall after x hours.
• The \(+6\) accounts for the snow already present.

By substituting \(x = 9\) (for the 9 hours the snow fell), we calculate that the new total snow depth is 10.5 inches after the storm has passed.

School Closure Conditions

Educational institutions often have guidelines to determine when weather conditions, such as heavy snowfall, make it unsafe to operate. In the Apple County School District, the policy states that if there is a foot or more of snow, schools will be closed. A foot is equivalent to 12 inches.

During the storm described in this problem, the total snowfall was calculated to be 10.5 inches. This value is below the school's cutoff of 12 inches. Since the snow depth did not reach a foot, the conditions necessary for school closure were not met. Decisions like these are crucial in ensuring student and staff safety, while also maintaining educational schedules when possible.

Graphical Representation of Equations

Visualizing equations on a graph can offer great insights. The linear equation \(y=\frac{1}{2}x+6\) describes how the total snow depth changes over time. Mapping this equation as a graph provides a clear picture of the snowfall's progression and how it impacts cumulative snow depth.

In this linear equation:

• The slope is \(\frac{1}{2}\), indicating a steady increase in snow depth by \(\frac{1}{2}\) inch every hour.
• The y-intercept is 6, showing the initial condition with 6 inches of snow already present before the storm began.

On a graph:

• The x-axis represents the hours of snowfall.
• The y-axis represents the total snow depth.

By plotting this graph, students can observe how the snow depth reaches 10.5 inches after 9 hours, but does not reach the threshold for school closure, providing a visual explanation along with the numerical calculation.