Let's analyze the equation [tex]\( y = \frac{1}{2}x + 8 \)[/tex] in the context of the given snowstorm scenario and interpret the slope and the y-intercept.
### Understanding the Equation
The equation [tex]\( y = \frac{1}{2}x + 8 \)[/tex] is a linear model representing the depth of snow [tex]\( y \)[/tex] on the ground after [tex]\( x \)[/tex] hours of snowfall.
### Slope Interpretation
The slope of the equation is [tex]\( \frac{1}{2} \)[/tex]. In this context, the slope represents the rate of snowfall.
- Meaning of the Slope: The slope [tex]\( \frac{1}{2} \)[/tex] indicates that the snow is accumulating at a rate of [tex]\( \frac{1}{2} \)[/tex] inch per hour. This means that for each hour that passes, the depth of the snow increases by [tex]\( \frac{1}{2} \)[/tex] inch.
### Y-Intercept Interpretation
The y-intercept of the equation is 8. The y-intercept is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
- Meaning of the Y-Intercept: The y-intercept 8 indicates the initial amount of snow on the ground before the snowfall began. This means that before any additional snow started falling, there were already 8 inches of snow on the ground.
### Total Snow Depth Calculation
Given that snow fell for 9 hours, we can calculate the total snow depth:
1. Determine the amount of snow accumulated over 9 hours:
- Snowfall rate: [tex]\( \frac{1}{2} \)[/tex] inch per hour
- Duration of snowfall: 9 hours
- Total accumulation due to snowfall: [tex]\( \frac{1}{2} \times 9 = 4.5 \)[/tex] inches
2. Add the initial snow depth:
- Initial snow depth: 8 inches
- Total snow depth after 9 hours: [tex]\( 8 + 4.5 = 12.5 \)[/tex] inches
### Conclusion
- The slope [tex]\( \frac{1}{2} \)[/tex] represents the rate at which snow is falling, which is [tex]\( \frac{1}{2} \)[/tex] inch per hour.
- The y-intercept 8 represents the initial snow depth before the snowfall began, which is 8 inches.
- After 9 hours of continuous snowfall at the given rate, the total depth of snow on the ground is 12.5 inches.
