SOLUTION: 3D application of trigonometric problem: Elena sees that the angle of elevation to the top of a tower, 80 m high, which is due south of her position is 37°. The top of another

Algebra ->  Trigonometry-basics -> SOLUTION: 3D application of trigonometric problem: Elena sees that the angle of elevation to the top of a tower, 80 m high, which is due south of her position is 37°. The top of another       Log On


   

Question 716067: 3D application of trigonometric problem:
Elena sees that the angle of elevation to the top of a tower, 80 m high, which is due south of her position is 37°. The top of another tower, 60 m high, which is due west of her is at an angle of elevation of 28°. Calculate the horizontal distance between the two towers.
I have attempted to construct a diagram to 'unfold' the triangles to lay them flat, to no success.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3D diagrams are hard to draw and even harder to describe how to draw them. So I am not going to try to tell you how to make a 3D diagram.

Let's name the key points:
Let E be where Elena is
Let A be the base/bottom of the 80m tower and B be the top of this tower.
Let X be the base/bottom of the 60m tower and Y be the top of this tower.

This makes the triangles: EAX, EAB and EXY. Triangle EAX, which is flat on the ground, is a right triangle because A is south of E and X is west of E making angle AEX a right triangle. Triangle EAB is a right triangle because the ground is horizontal and the tower is vertical making angle EAB a right angle. The same reasoning applies to triangle EXY.

The horizontal distance between the towers will the the length of segment AX. This is the hypotenuse of triangle EAX. To find AX we will need either the other two sides of the triangle or an acute angle and a side. Currently we only know that angle AEX is a right triangle. But sides EA and EX are each sides of one of the other right triangles. And we know an acute angle and a side in both triangle of those other triangles. So we can solve for EA using triangle EAB and solve for EX using triangle EXY and then use those values to find AX.

In triangle EAB we know that the angle of elevation, angle AEB, is 28 degrees and we know the height of the tower, 80. Since we are looking for side EA and since EA is adjacent to angle EAB and since the height of the tower is opposite to angle EAB, we will use tan:
tan%2837%29+=+80%2F%28EA%29
Multiplying both sides by (EA):
%28EA%29%2Atan%2837%29+=+80
Dividing both sides by tan(37):
%28EA%29+=+80%2Ftan%2837%29
Using our calculators:
%28EA%29+=+106.16358572963280297275780592695

Using the same logic in triangle EXY we get:
tan%2828%29+=+60%2F%28EX%29
Multiplying both sides by (EX):
%28EX%29%2Atan%2828%29+=+60
Dividing both sides by tan(28):
%28EX%29+=+60%2Ftan%2828%29
Using our calculators:
%28EX%29+=+112.84358792077992074165025574976

Now that we have EA and EX we can use the Pythagorean Theorem to find AX:
%28EA%29%5E2%2B%28EX%29%5E2=%28AX%29%5E2
or
sqrt%28%28EA%29%5E2%2B%28EX%29%5E2%29=%28AX%29
I will leave it up to you to finish:
  1. Decide how many decimal places to round off EA and EX (or use all of them if you want to type that much).
  2. Substitute these numbers for EA and EX into the equation sqrt%28%28EA%29%5E2%2B%28EX%29%5E2%29=%28AX%29
  3. Use your calculator to find AX.