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A portion of a truss vehicle bridge has steel

beams that form an isosceles triangle with the
dimensions shown. A pedestrian handrail is
attached to the side of the bridge so that it is
parallel to the road.
A
24 feet
28.5 feet
30 feet
What is the height, x, of the handrail above th
road to the nearest tenth of a foot?

A portion of a truss vehicle bridge has steel beams that form an isosceles triangle-example-1

1 Answer

3 votes

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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A portion of a truss vehicle bridge has steel beams that form an isosceles triangle-example-1
User Tony Dinh
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