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When Patty wakes up, there are 6 inches of snow on the ground. As the snow continues to fall, the depth of the snow increases by 2 inches per hour.



Write an equation that shows the relationship between x, the number of hours after Patty wakes up, and y, the depth of the snow in inches.



y =

Verified Answer :

  • The problem describes a linear relationship with an initial value and a constant rate of growth.
  • The initial depth of snow at \(x=0\) is \(b=6\).
  • The rate of change in snow depth is \(m=2\) per hour.
  • Using the slope-intercept form \(y=mx+b\), the equation is \(\boxed{y = 2x + 6}\).

Explanation

  1. Analyzing the problem parameters

We are given that when Patty wakes up at \(x = 0\) hours, the snow depth \(y\) is \(6\) inches. This represents the initial value of the snow, which corresponds to the y-intercept of the linear relationship. As time progresses, the snow continues to fall at a constant rate of \(2\) inches per hour. This rate of increase represents the slope of the linear equation. Therefore, for every hour \(x\) that passes, the depth increases by \(2x\) inches, and we must add the initial depth of \(6\) inches to find the total depth \(y\).

  1. Formulating the linear equation

A linear equation in slope-intercept form is given by \(y = mx + b\), where \(m\) is the rate of change (slope) and \(b\) is the initial value (y-intercept).
From our analysis:

  • The rate of change \(m\) is \(2\) inches per hour.
  • The initial value \(b\) is \(6\) inches.

Substituting these values into the slope-intercept form gives us:
\[y = 2x + 6\]
This equation models the total depth of the snow \(y\) after \(x\) hours.

Examples

Linear relationships like this are essential for predicting future values based on constant rates of change. For example, if you are filling a pool with water at a steady rate, or tracking your savings account balance with a fixed monthly deposit, you are using the same principle. Understanding the components of a linear equation allows you to model real-world scenarios where growth is consistent, helping you plan and forecast effectively.