Algebra 2 Word problem Question

Question

You want to measure the height of a mountain. You are at an unknown distance from a mountain. The angle of elevation to the top of the mountain is 65 degrees. You step back 100 yards and measure the angle of elevation to be 60 degrees. Find the heights of the mountain and your original distance from the mountain. 

Answer 1

Call the top of the mountain M, the first spot you measure from F and the second spot you measure from S.  That gives you a triangle, ΔMFS.

 

You can see if you draw the diagram that the angle of elevation from F is supplementary to angle MFS.  That means that angle MFS is 180 - 65 = 115°.  You also know that angle FSM is the angle of elevation from spot S, and that's 60°.  That leaves 5° for angle FMS.

 

Now you can use the law of sines to find the distance from F to the top of the mountain, which is the length of FM.

 

sin 5° / 100 = sin 60° / FM

FM = 100 sin 60° / sin 5° = 993.65 yards

 

Now call B the point on the ground directly below the top of the mountain.construct another triangle.  Construct another triangle ΔMFB.  We already know that FM = 993.65 yards, and angle BFM is 65°.  That means that the height of the mountain, MB, divided by FM (which is the hypotenuse of ΔMFB) is sin 65°.

 

MB / FM = sin 65°

MB = FM sin 65°

MB = 993.65 * 0.906 = 900.55 yards.  That's the height of the mountain.

 

Finally,

 

BF / FM = cos 65°

BF = FM cos 65°

BF = 993.65 * 0.423 = 419.93.  That's the original distance from the center of the base of the mountain.

 

 

Answer 2

To solve this equation you need to set up two right triangles. one with the base being the length of the initial distance x and one with the base being x+100. Both triangles will have height y which is the height of the mountain. From here you can set up  a system of equations because tangent= opposite/ adjacent and the angle of elevation in the first triangle is 65 degrees you can say tan(65) = y/x. Similarly you can set up the equation for the second triangle tan(60)= y/(100+x).

At this point you have a system of equations and can solve for the variables. To do this you can either put both equations into the terms of one  of the variables and set them equal to each other or put one of the equations in terms of one of the variables and substitute it in the other equation (which are both actually doing the same thing just in slightly different orders. When I plugged in the numbers I got that x  was approximately 419.93578 yards and y was about 900.555 yards.

Hope this helps.