To find the slopes of the top rail \(f(x)\) and the bottom rail \(g(x)\), we will need to calculate the slope using the formula for slope:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
where \(y\) represents the distance from the floor, and \(x\) represents the step number.
Given Data:
-
Top rail: \(f(x)\)
- When \(x = 1\), \(f(1) = 41\)
- When \(x = 2\), \(f(2) = 48\)
- When \(x = 3\), \(f(3) = 55\)
-
Bottom rail: \(g(x)\)
- When \(x = 1\), \(g(1) = 9\)
- When \(x = 2\), \(g(2) = 16\)
- When \(x = 3\), \(g(3) = 23\)
Step 1: Calculate the slope of the top rail \(f(x)\)
Using the first two points of the top rail:
\[ \text{slope of } f = \frac{f(2) - f(1)}{2 - 1} = \frac{48 - 41}{2 - 1} = \frac{7}{1} = 7 \]
Step 2: Calculate the slope of the bottom rail \(g(x)\)
Using the first two points of the bottom rail:
\[ \text{slope of } g = \frac{g(2) - g(1)}{2 - 1} = \frac{16 - 9}{2 - 1} = \frac{7}{1} = 7 \]
Conclusion
Both slopes are equal:
- The slope of the top rail \(f(x)\) is \(7\).
- The slope of the bottom rail \(g(x)\) is \(7\).
Since the slopes of both rails are equal, we can conclude that the top and bottom rails are parallel.
Final Answer
B. Since both rails have a slope of \(7\), the top and bottom rails are parallel.