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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.)

1 answer

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.