Let's solve the problem step-by-step.
1. Understanding the Real-World Problem:
- Caitlin's railing starts at a height of 36 inches at the top of the stairs.
- The height of the railing decreases as we move horizontally along the stairs.
- Specifically, the railing decreases by 9 inches for every 12 inches we move horizontally.
2. Determining the Slope:
- The slope (m) of a line can be found using the formula [tex]\( \text{slope} = \frac{\text{rise}}{\text{run}} \)[/tex].
- Here, the "rise" is negative because the height of the railing decreases. So, the rise is -9 inches, and the run (horizontal distance) is 12 inches.
[tex]\[
\text{slope} (m) = \frac{-9}{12} = -\frac{3}{4}
\][/tex]
3. Forming the Linear Equation:
- The general form of a linear equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept (the starting height of the railing in this case).
- From the given problem, the y-intercept [tex]\( b \)[/tex] is 36 inches because this is the initial height of the railing.
- The slope [tex]\( m \)[/tex] has been calculated as [tex]\( -\frac{3}{4} \)[/tex].
4. Writing the Equation:
- Substituting the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the slope-intercept form equation, we get:
[tex]\[
y = -\frac{3}{4} x + 36
\][/tex]
Thus, the function that represents the height [tex]\( y \)[/tex] of the railing in inches according to the horizontal distance [tex]\( x \)[/tex] from the top of the stairs is:
[tex]\[
y = -\frac{3}{4} x + 36
\][/tex]
Answer:
[tex]\[ y = -\frac{3}{4} x + 36 \][/tex]
