Astuntman uses a 30 foot rope to swing 136 degrees between two platforms of equal height, grazing the ground in the middle of the swing. if the rope stays taut throughout the swing, how far above the ground was the stuntman at the beginning and the end of the swing?

Alright, let's start with what we know in this equation. If the two platforms are of equal height, and the stuntman swings 136 degrees on his rope to reach them, we should be able to split up his swing into two equal triangles which both have the angle on top equal to 68 degrees. 

Another thing we can learn from the question is the measurements of the other two angles in both of the triangles. If the 30 feet of rope is taut throughout their swing, we know that two of the triangle's sides are 30 ft, and if a triangle has two equal sides, the anlgles opposite of those sides should have the same measurements. To find those measurements, what we need to do is take the sum of the leftover angles, which is 180-68, or 112 degrees, and then divide that by two. So, the other angles in both triangles should both be 56 degrees.

Our next step should be to use the Law of Sines to find the measurement of the third side of the triangles. The law of Sines is the idea that sin(a)/A = sin(b)/B = sin(c)/C where the lowercase letters represent an angle and the uppercase letters represent the side opposing the angle of the same letter. Using this, we can take the top angle and the side we don't know and set it equal to one of the other sides. So our equation should look something like sin(68)/x = sin(56)/30. Next we need to cross multiply, giving us sin(68)*30 = sin(56)*x. Simplifying this should give us 27.815 = sin(56)*x, and when we divide both sides by sin(56) we should end up with a measurement of about 33.551 for the third side of our triangle.

Once we have that information, we need to set up another triangle that connects the ground to one of the platforms, with the hypotenuse being the last measurement of 33.551. This triangle should make a right angle of 90 degrees between the ground and the platform, meaning we only have to find two more angles. To do this, we can look at the angle where the ground connected with the rope in the first part. We found that this angle is 56 degrees, and this angle is complementary to the one that we are trying to find in our new triangle, which gives a good place to start. Complementary angles add up to 90 degrees, so to find the new angle's measurement, all we have to do is subtract 56 from 90, which gives us 44 degrees as the measurement of our new angle.

Next, we just have to find the last angle's measurement, which should be pretty easy once we know the other two angles. Because all the angles in a triangle add up to 180 degrees, we just have to subtract the two angles we know from 180! 180-44-90= 46, which should be our last angle's measurement. Now that we have the measurement of one side and all the angles, we can use the Law of Sines again to find out the height from the ground to one of the platforms. To do this, we need to set up a proportion again, and this time it should look something like this: sin(90)/33.551 = sin(46)/x. Cross multiplying will give us sin(90)*x = sin(46)*33.551, and before we simplify, it's good to remember that sin(90) is the same thing as 1, so that makes this last step a little easier. After remembering that, simplifying gives us x = 30.255, which should be the height from the ground to either one of the platforms.