Use the Pythagorean triangle to solve for the third side of the triangle.

Simplify and you have the answer:

This is a typical angle of elevation problem. 

From the question, we can infer that the ratio of the mountain height to the surveyor's horizontal distance from the mountain is equal to the tangent of 35 degrees - the mountain height is the "opposite side", while the horizontal distance is the "adjacent side".  In other words:

Solving for the mountain height gives a vertical distance of 1680.5 feet.

Here, we can use the Pythagorean Theorem.

This states that the sum of the squares of the legs is equal to the square of the hypotenuse.

Thus, we plug into the formula to get our hypotenuse. 

The altitude of the triangle splits the 60 degree angle into two 30 degree angles.  Notice that this will create two right triangles with 30-60-90 angles.  This indicates that the full triangle is an equilateral triangle.

The altitude represents the adjacent side of the split triangles. The bisected angle is 30 degrees.  Find the hypotenuse of either split triangle.  This is the side length of the equilateral triangle. 

Rationalize the denominator.

Since the full triangle has three sides, multiply this value by three for the perimeter.

The question is looking for the length of the ramp which would be the length of the part one would walk on and is also synonymous with the hypotenuse of our right triangle. With the given angle and opposite side we can use the sine function to solve for the length of the ramp because 

.

We have all but the length of the hypotenuse.

Solve for the hypotenuse:

Plug in the angles and sides given:

Concerning the angle of elevation which we are to find, we know the side length opposite this angle and the side length adjacent to it.

We use

.

Take the inverse of tan to find the corresponding angle:

A short cut for solving is to realize right triangles with equal sides must have two 45° angles!

Step 1: Recall the Pythagorean Theorem:

, where  are the legs of the triangle.

Step 2: Plug in the values that are given for the legs into the theorem:



Evaluate:



Add:



Step 3: To find , take the square root of both sides:



Reduce (if possible):



The length of the hypotenuse is .

Step 1: To find the missing side of a right triangle using the Pythagorean Theorem: 

.

Step 2: Substitute in the known values.

Step 3: Solve for the missing variable by manipulating the equation to isolate the variable.

Recall the Law of Sines for a generic triangle, as shown above:

Plug in the given values into the Law of Sines:

Rearrange the equation to solve for :

Make sure to round to two places after the decimal.

Recall the Law of Sines:

Plug in the given values into the Law of Sines:

Rearrange the equation to solve for :

Make sure to round to two places after the decimal.