UpStudy Free Solution:

To calculate the vertical distance between the two stations, we can use the Pythagorean theorem. The slope distance is the hypotenuse of a right triangle, and the horizontal distance is one of the legs. The vertical distance is the other leg of the triangle.

Let \(d\) be the slope distance, \(h\) be the horizontal distance, and \(v\) be the vertical distance. According to the Pythagorean theorem:

\[d^ 2 = h^ 2 + v^ 2\]

Solving for \(v\):

\[v = \sqrt { d^ 2 - h^ 2} \]

Given:

\[d = 125.444 \space \text { m} \]

\[h = 122.967 \space \text { m} \]

Plugging in the values:

\[v = \sqrt { 125.444^ 2 - 122.967^ 2} \]

First, calculate the squares:

\[125.444^ 2 = 15741.111936\]

\[122.967^ 2 = 15120.023289\]

Subtract the squares:

\[15741.111936 - 15120.023289 = 621.088647\]

Now, take the square root:

\[v = \sqrt { 621.088647} \approx 24.928\]

So, the vertical distance is approximately \(24.928\) meters.

Answer: 24.928

Supplemental Knowledge:

To calculate the vertical distance between two points when the slope distance and horizontal distance are known, we can use the Pythagorean theorem. The relationship between the slope distance (hypotenuse), horizontal distance (base), and vertical distance (height) in a right triangle is given by:

\[\text { slope distance} ^ 2 = \text { horizontal distance} ^ 2 + \text { vertical distance} ^ 2\]

Rearranging to solve for the vertical distance (\(v\)):

\[v = \sqrt { \text { slope distance} ^ 2 - \text { horizontal distance} ^ 2} \]

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