A climber, working his way up a vertical cliff stops for a rest on the ledge. The ledge where he stopped is 12 meters wude. The angle of depression from the edge of the ledge to the bottom of the cliff is 72 degree and the angle of elevation to the top of the cliff is 88 degree. Find the height of the cliff.
If the cliff is vertical, how can there be a ledge? And how can the angle looking down be other than 90 degrees?
I don't get it.
It appears that the situation looks like a shelf sticking out from a vertical wall. The shelf is 12m wide, so we have two right triangles:
Looking down, from shelf height h, we have
h/12 = tan 72
Looking up to the peak at height p above the shelf, we have
p/12 = tan 88
The total cliff elevation is h+p
To find the height of the cliff, we can use trigonometry and the given information.
Let's consider the triangle formed by the climber, the bottom of the cliff, and the top of the cliff.
We can start by finding the length of the base of the triangle, which is the distance from the edge of the ledge to the bottom of the cliff. This length can be found by using the tangent of the angle of depression.
Tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side, so we have:
tan(72°) = opposite/adjacent
The adjacent side is 12 meters (the width of the ledge), and we want to find the opposite side. Rearranging the equation, we get:
opposite = tan(72°) * adjacent
= tan(72°) * 12
Using a calculator, we can calculate:
opposite ≈ 48.271 meters
Now, we need to find the length of the other side of the triangle, which is the height of the cliff. This can be found using the tangent of the angle of elevation.
tan(88°) = opposite/adjacent
The opposite side is the height of the cliff, and the adjacent side is the same 12 meters. We can rearrange the equation to solve for the height:
height = tan(88°) * 12
Using a calculator, we find:
height ≈ 130.216 meters
Therefore, the height of the cliff is approximately 130.216 meters.