Question

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Part 1
A hand rail is installed along the stairs of a new house as shown in the figure. The table shows the​ distance, in​ inches, of the top rail​ f(x) and bottom rail​ g(x) from the floor for the middle of each numbered step x. Determine the slope of each rail. Are the top and bottom rails​ parallel?
x
f left parenthesis x right parenthesisf(x)
g left parenthesis x right parenthesisg(x)
1
4141
99
2
4848
1616
3
5555
2323
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Part 1
Select the correct choice below and fill in the answer​ box(es) to complete your choice.
A.
The slope of the top rail is enter your response here​, and the slope of the bottom rail is enter your response here. Since the product of their slopes is not minus−​1, the top and bottom rails are not parallel.
​(Simplify your​ answers.)
B.
Since both rails have a slope of enter your response here​, the top and bottom rails are parallel.
​(Simplify your​ answer.)
C.
The slope of the top rail is enter your response here​, and the slope of the bottom rail is enter your response here. Since the slopes are​ different, the top and bottom rails are not parallel.
​(Simplify your​ answers.)
D.
The slope of the top rail is enter your response here​, and the slope of the bottom rail is enter your response here. Since the product of their slopes is minus−​1, the top and bottom rails are parallel.
​(Simplify your​ answers.)

Answer 1

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.**