Question

two lanes of a race track run parallel to each other. the outer lane is represented by the equation y = 0.5x - 2. the inner lane passes through the point (4,0). 22) what is the slope of the inner lane? explain your reasoning. 23) write the equation in slope - intercept form for the inner lane. 24) a safety barrier is to be placed at a right angle to the inner lane at the point (4,0). write the equation of the line representing this safety barrier.

Explanation:

Step1: Recall parallel - line property

Parallel lines have the same slope. The outer - lane equation is $y = 0.5x−2$, which is in slope - intercept form $y=mx + b$ where $m$ is the slope. So the slope of the inner lane is also $0.5$ since the two lanes are parallel.

Step2: Find the y - intercept of the inner lane

We know the slope $m = 0.5$ and the line passes through the point $(4,0)$. Substitute $x = 4$, $y = 0$ and $m=0.5$ into $y=mx + b$. We get $0=0.5\times4 + b$. Solving for $b$:
\[

$$\begin{align*} 0&=2 + b\\ b&=- 2 \end{align*}$$

\]
The equation of the inner lane in slope - intercept form is $y = 0.5x-2$.

Step3: Find the slope of the perpendicular line

The slope of a line perpendicular to a line with slope $m$ is $-\frac{1}{m}$. Since the slope of the inner lane is $m = 0.5=\frac{1}{2}$, the slope of the safety - barrier line (perpendicular to the inner lane) is $m'=-2$.

Step4: Find the equation of the safety - barrier line

We know the slope $m'=-2$ and the line passes through the point $(4,0)$. Substitute into the point - slope form $y - y_1=m'(x - x_1)$ (where $(x_1,y_1)=(4,0)$). We get $y-0=-2(x - 4)$. Simplifying gives $y=-2x + 8$.

Answer:

1. The slope of the inner lane is $0.5$ because parallel lines have the same slope and the outer - lane has a slope of $0.5$.
2. The equation of the inner lane in slope - intercept form is $y = 0.5x-2$.
3. The equation of the line representing the safety barrier is $y=-2x + 8$.