Answer :
To determine the correct function that represents the height [tex]\( y \)[/tex] of the railing in inches according to the horizontal distance [tex]\( x \)[/tex] in inches from the top of the stairs, follow these steps:
### Step-by-Step Solution
1. Identify Key Pieces of Information:
- The railing starts at a height of 36 inches.
- The height decreases by 9 inches for every 12 horizontal inches.
2. Determine the Slope:
- The slope [tex]\( m \)[/tex] of a line is given by the change in vertical distance divided by the change in horizontal distance.
- Here, the vertical decrease is 9 inches and the horizontal distance is 12 inches.
[tex]\[ m = \frac{\text{change in height}}{\text{horizontal distance}} = \frac{-9}{12} = -\frac{3}{4} \][/tex]
- The negative sign indicates that the height is decreasing as you move horizontally.
3. Identify the Y-Intercept:
- The y-intercept ([tex]\( b \)[/tex]) is the initial height of the railing when [tex]\( x = 0 \)[/tex].
- In this problem, the railing starts at a height of 36 inches, so [tex]\( b = 36 \)[/tex].
4. Construct the Function:
- Using the slope-intercept form of a linear equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept:
[tex]\[ y = -\frac{3}{4}x + 36 \][/tex]
5. Verify the Options:
- Now, let's check which of the given options corresponds to this function:
- [tex]\( y = -\frac{3}{4} x + 36 \)[/tex]
- [tex]\( y = -3 x + 36 \)[/tex]
- [tex]\( y = \frac{3}{4} x + 36 \)[/tex]
- [tex]\( y = 3 x + 36 \)[/tex]
The correct function that represents the height [tex]\( y \)[/tex] of the railing in inches according to the horizontal distance [tex]\( x \)[/tex] is:
[tex]\[ y = -\frac{3}{4} x + 36 \][/tex]
### Conclusion
Therefore, the correct function to represent the height of the railing is:
[tex]\[ y = -\frac{3}{4} x + 36 \][/tex]
### Step-by-Step Solution
1. Identify Key Pieces of Information:
- The railing starts at a height of 36 inches.
- The height decreases by 9 inches for every 12 horizontal inches.
2. Determine the Slope:
- The slope [tex]\( m \)[/tex] of a line is given by the change in vertical distance divided by the change in horizontal distance.
- Here, the vertical decrease is 9 inches and the horizontal distance is 12 inches.
[tex]\[ m = \frac{\text{change in height}}{\text{horizontal distance}} = \frac{-9}{12} = -\frac{3}{4} \][/tex]
- The negative sign indicates that the height is decreasing as you move horizontally.
3. Identify the Y-Intercept:
- The y-intercept ([tex]\( b \)[/tex]) is the initial height of the railing when [tex]\( x = 0 \)[/tex].
- In this problem, the railing starts at a height of 36 inches, so [tex]\( b = 36 \)[/tex].
4. Construct the Function:
- Using the slope-intercept form of a linear equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept:
[tex]\[ y = -\frac{3}{4}x + 36 \][/tex]
5. Verify the Options:
- Now, let's check which of the given options corresponds to this function:
- [tex]\( y = -\frac{3}{4} x + 36 \)[/tex]
- [tex]\( y = -3 x + 36 \)[/tex]
- [tex]\( y = \frac{3}{4} x + 36 \)[/tex]
- [tex]\( y = 3 x + 36 \)[/tex]
The correct function that represents the height [tex]\( y \)[/tex] of the railing in inches according to the horizontal distance [tex]\( x \)[/tex] is:
[tex]\[ y = -\frac{3}{4} x + 36 \][/tex]
### Conclusion
Therefore, the correct function to represent the height of the railing is:
[tex]\[ y = -\frac{3}{4} x + 36 \][/tex]
