Answer:
3.5 feet
Explanation:
You want the difference in height between similar isosceles triangles, one with height of 30 ft and a base of 24 ft, the other with a side length of 28.5 ft.
Relations
We can find the side length of the larger triangle using the Pythagorean theorem. It will be ...
longer side = √(30² +12²) ≈ 32.311 ft
Similar triangles
Then the length x is the difference between the altitudes of the triangles. The altitudes are proportional to the side lengths, so we have ...
(30 -x)/28.5 = 30/32.311
x = 30-(28.5)(30/32.311) = 30(1 -28.5/32.311) ≈ 3.538 ≈ 3.5 . . . . feet
The hand rail is about 3.5 feet above the bridge deck.
Trigonometry
We recognize that the distance from the hand rail to the top of the triangle is the product of the given side length (28.5 ft) and the cosine of the angle between the side and the altitude.
The tangent of that angle is the ratio of its opposite side (12 ft) to its adjacent side (30 ft), or θ = arctan(12/30).
The value of x is the difference of the altitudes of the triangles, so is ...
x = 30 -28.5·cos(arctan(12/30)) ≈ 3.5 ft
We find this easier to compute.
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