Question

6. A hiker looks down into a valley with binoculars. The angle of depression to the farthest edge of the river is 61. The angle to the closest edge of the river below is 63. If the valley is 1250

feet deep, how wide is the river? Round the answer to the nearest foot.

Answer 1

AO/OD. ⇒ √3 = 10/x ⇒ x = 10/√3 units

Explanation:

Solution: In the figure given above, there are two angles of depression formed from point A, which are ∠BAC=30° and ∠DAC=60°. We know that, ∠DAC=∠ADO (by using alternate interior angles theorem ). So, ∠ADO=60°. By applying the angle of depression formula in AOD, we get tan 60° = AO/OD. ⇒ √3 = 10/x ⇒ x = 10/√3 units

Answer 2

w ≈ 640 feet

Explanation:

Let's call the distance from the hiker's position to the closest edge of the river "x", and the width of the river "w". We can use trigonometry to set up two equations involving these values:

In the first triangle, the angle of depression is 63 degrees, and the opposite side is x + w. Therefore, we can use the tangent function:

tan(63) = (x + w) / 1250

In the second triangle, the angle of depression is 61 degrees, and the opposite side is x. Therefore, we can use the tangent function again:

tan(61) = x / 1250

Now we can solve these equations for "w" and "x", respectively:

w = 1250 * tan(63) - x

x = 1250 * tan(61)

Substituting the second equation into the first:

w = 1250 * tan(63) - 1250 * tan(61)

Plugging this into a calculator, we get:

w ≈ 640 feet

Therefore, the width of the river is approximately 640 feet rounded to the nearest foot.