R and S are in the same horizontal plane as the base of a cliff, but R is 132.6 meters nearer it than is S. They find the angles of elevation of the top of the cliff to be 32 degrees and 36 minutes and 63 degrees and 13 minutes. how high is the cliff?

Set up your diagram. If x is the distance of R from the cliff, and h is the cliff height,

S: h/(x+132.6) = tan 32°36' = 0.6395
R: h/x = tan 63°13' = 1.9811

Since h is the same in both equations,

.6395(x+132.6) = 1.9811x

.6395x + 84.7977 = 1.9811x
x = 63.2064

So, h = 63.2064 * 1.9811 = 125.2119

To find the height of the cliff, we can use trigonometry and the given information about the angles of elevation and the distance between R and S.

Let's break down the problem step by step:

Step 1: Determine the distance between R and S.
Given that R is 132.6 meters nearer to the base of the cliff than S, we can calculate the distance between R and S as follows:
Distance between R and S = 132.6 meters.

Step 2: Draw a diagram.
Sketch a diagram representing the situation described in the problem. Label the base of the cliff as B, the top of the cliff as T, and the positions of R and S accordingly.

Step 3: Identify the right-angled triangles.
There are two right-angled triangles formed in this problem. One triangle includes the positions of R, T, and the base of the cliff (B). The other triangle includes the positions of S, T, and B.

Step 4: Use trigonometry to find the height of the cliff.
In the triangle with R, T, and B:
tan(angle of elevation R) = height of cliff / distance between R and B

In the triangle with S, T, and B:
tan(angle of elevation S) = height of cliff / distance between S and B

We need to convert the angles from degrees and minutes to decimal degrees before using the tangent function. Let's do that:
Angle of elevation R = 32 degrees + 36 minutes = 32 + 36/60 = 32.6 degrees
Angle of elevation S = 63 degrees + 13 minutes = 63 + 13/60 = 63.22 degrees

Now we can use trigonometry to find the height of the cliff:
tan(32.6 degrees) = height of cliff / 132.6 meters
tan(63.22 degrees) = height of cliff / distance between S and B

Step 5: Solve the equations.
Using a scientific calculator or trigonometric tables, calculate the values of the tangent function for the given angles. Solve the equations simultaneously to find the height of the cliff.

After following these steps, you should be able to determine the height of the cliff.