To determine the equation of a line that models [tex]\( y \)[/tex], the number of miles Dudley has left to travel after biking [tex]\( x \)[/tex] hours, we need to establish a relationship between the distance he travels over time.
We know the following:
1. Total Distance to Cover: Dudley wants to bike a total of 40 miles.
2. Rate of Travel: He travels 30 miles every 2 hours.
First, we need to find Dudley's rate of travel in miles per hour. Given that he travels 30 miles in 2 hours, his rate of travel is:
[tex]\[
\text{Rate per hour} = \frac{30 \, \text{miles}}{2 \, \text{hours}} = 15 \, \text{miles per hour}
\][/tex]
Next, let's establish an equation for the distance remaining after [tex]\( x \)[/tex] hours. Initially, he plans to travel 40 miles. The distance Dudley travels after [tex]\( x \)[/tex] hours can be calculated as:
[tex]\[
\text{Distance traveled} = 15 \, \text{miles per hour} \times x \, \text{hours} = 15x
\][/tex]
The distance he has left to travel [tex]\( y \)[/tex] is the total distance minus the distance he has already traveled:
[tex]\[
y = 40 \, \text{miles} - 15x \, \text{miles}
\][/tex]
Therefore, the equation that models the number of miles left to travel after [tex]\( x \)[/tex] hours is:
[tex]\[
y = -15x + 40
\][/tex]
Given the choices:
A. [tex]\( y = -15x - 40 \)[/tex]
B. [tex]\( y = -30x + 40 \)[/tex]
C. [tex]\( y = -15x + 40 \)[/tex]
D. [tex]\( y = 30x - 40 \)[/tex]
The correct equation is:
C. [tex]\( y = -15x + 40 \)[/tex]
