To determine which graph and equation accurately represent Emily's distance from home, [tex]\( D \)[/tex], after traveling for [tex]\( h \)[/tex] hours, let's analyze the problem step by step.
### Problem Breakdown
1. Initial Distance: Emily starts 280 miles away from home.
2. Rate of Travel: Emily is traveling back home at a constant speed of 40 miles per hour.
### Formulating the Equation
- When [tex]\( h = 0 \)[/tex]:
Emily has not started traveling yet, so the distance [tex]\( D \)[/tex] will be 280 miles.
- When Emily travels [tex]\( h \)[/tex] hours, she will cover a distance of [tex]\( 40 \times h \)[/tex] miles.
- This distance [tex]\( 40h \)[/tex] should be subtracted from the initial distance of 280 miles to get the current distance from home.
Thus, the equation becomes:
[tex]\[ D = \text{initial distance} - (\text{speed} \times \text{time}) \][/tex]
[tex]\[ D = 280 - 40h \][/tex]
### Checking the Options
From the given options:
- [tex]\( D = 280 h - 40 \)[/tex]
- [tex]\( D = 280 - 40 h \)[/tex] [tex]\( \leftarrow \text{This one matches our equation}\right)
- \( D = (280 - 40) h \)[/tex]
- [tex]\( D = 40 h - 280 \)[/tex]
The correct equation is:
[tex]\[ D = 280 - 40h \][/tex]
### Graph
To select the correct graph, it should represent a linear relationship where:
- The [tex]\( y \)[/tex]-intercept (when [tex]\( h = 0 \)[/tex]) is 280 miles.
- The slope of the line is negative because the distance decreases as time increases, which corresponds to a decrease of 40 miles per hour.
So, the correct equation representing Emily's distance from home after traveling [tex]\( h \)[/tex] hours is:
[tex]\[ D = 280 - 40h \][/tex]
And the graph that represents this equation would start at 280 on the y-axis and slope downward with a gradient of -40. This indicates that each hour traveled decreases the distance by 40 miles.
