Question

From the top of a canyon, the angle of depression to the far side of the river is 52°, and the angle of depression to the near side of the river is 71°. The depth of the canyon is 230 feet. What is the width of the river at the bottom of the canyon? Round to the nearest foot.

Trig question

Answer 1

49ft

Explanation:

Sketching out the problem, we get something like the diagram below. Assuming the canyon makes a roughly 90° angle with the ground, we can see that we have two right-angled triangles formed.

Focusing on the first triangle: since we have the angle, and we know the height of the canyon is 230ft, we can also find the combined width of the ground and river (let's call this y) using cos:

cos38=230/y

0.7880107536=230/y

y=230/0.7880107536= 291.874189467ft

Focusing now on the first triangle: since we have the angle, and we know the height of the canyon is 230ft, we can find the width of the ground (let's call this x) using cos:

cos19=230/x

0.94551857559=230/x

x=230/0.94551857559= 243.252756673ft

Therefore, the width of the river must be the difference between the combined width of the ground and river, minus the width of the ground:

291.874189467ft - 243.252756673ft = 48.621432794ft ≈ 49ft

Answer 2

To solve this problem, we can use the law of tangents.

Let's call the width of the river at the bottom of the canyon "x". Then we can use the following two equations:

tan(52°) = 230 / (x/2)

tan(71°) = 230 / (x/2 + 230)

Solving for x:

x/2 = 230 / tan(52°)

x/2 + 230 = 230 / tan(71°)

Solving for x by substituting the first equation into the second:

x/2 + 230 = 230 / tan(71°) = 230 / tan(52°) * tan(71°) / tan(52°) = x/2 * tan(71°) / tan(52°)

x/2 * tan(52°) + 230 * tan(52°) = x/2 * tan(71°)

x * tan(52°) = 230 * tan(71°) - 230 * tan(52°)

x = (230 * tan(71°) - 230 * tan(52°)) / tan(52°)

Using a calculator, we find that x = 751.71 feet, so rounding to the nearest foot, the width of the river at the bottom of the canyon is 752 feet.