Trigonometry word problem

Question

I have a question in my homework and it asks this, and I’m confused on what the question is asking and how to start. Here’s the question: A bridge is going to be built across a river. To determine the width of the river, a surveyor on one bank sights the top of a pole, which is 3 m high, on the opposite bank. His optical device is mounted 1.2 m above the ground. The angle of elevation to the top of the pole is 8.5°. How wide is the river?

If anyone could help explain it a bit I would really appreciate it!!

Answer 1

Comment: Going into this solution, I can see why the wording made this heavily appear like it's difficult but once you see how it goes together; you'll see it for what it really is, a strangely phrased, simple problem:

Given:

H = 3 m

H₀ = 1.2 m

φ = 8.5°

Find:

W = ? m

Solution:

Looking along the river, with each side of the river expressing the width of it.

I suggest putting the optical device on the left side of the river and the pole on the right side of the river.

To whatever scale that you choose (if you make the diagram to scale), it'll be 1.2 m high on the left and 3.0 m high on the right.

The Angle of Elevation means there's a line connecting the top of the 1.2 m line to the top of the 3.0 m line and the angle it makes with the horizontal (which ends up being perpendicular to the 3.0 m line if you draw the line across from the 1.2 m line to touch the 3.0 m line, hint hint) is the angle φ

What should begin to form is a right triangle with a base of the width of the river, W, another base with the length of y (currently a dummy variable until we can relate it back to our givens) and the angle adjacent to the width and opposite of y is φ.

If you were to "cut out" the lengths of H₀, y, and H and put them in combinations to one another, you'd come to find out:

y + H₀ = H

∴ y = H - H₀

Using the properties of SOHCAHTOA, you'll find that:

tan φ = y / W

Solving this for W yields:

W = ( H - H₀ ) / tan φ = ( 3.0 - 1.2 ) / tan( 8.5° )

∴ W = 1.8 cot( 8.5° ) ≈ 12.04 m

I hope this helps! Message me in the comments if you have any questions, comments, or concerns!

If you'd like it, I can make a graph of the situation on Desmos. You need only ask.

Answer 2

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Above, imagine the left vertical is the 3m pole, and the right vertical is the stand for the optical device, with the Ws as the river.

Imagine a triangle with

-one vertex (C) at the optical device

-one vertex (A) on the top of the pole across the river

-one vertex (B) on the pole, level with the optical device

The angle at B measures 90º, since we chose that point to be level with the optical device (and we assume the pole is vertical).

The angle at C measures 8.5º (that's what 'angle of elevation' means).

The measure of the angle at A can now be calculated, thought we don't need it.

So side AB runs along an upper segment of the pole, and BC crosses the river horizontally. This gives the width of the river, which is what we're looking for. We know A is 3m off the ground, and B is 1.2m off the ground (since it's level with the optical device), so AB measures 3-1.2=1.8m.

Now we know an angle of the triangle (8.5º) and the side opposite that angle (1.8m), and we're looking for the side adjacent to the angle, BC. The trig function that relates Opposite and Adjacent sides is the tangent function, so we have that tan(8.5º)=1.8/BC. Solve for BC, and that's your answer.