Trigonometry Question

Question

Suppose you are looking at Taipei 101 (a tall building in Taipei, Taiwan) which has a height of about 509 meters. From where you are, you estimate the angle of elevation to the top of this building to be about 10 degrees.

How far are you from the building? Round to the nearest tenth: Answer

If the angle of elevation is only correct to the nearest three degrees, give the minimum and maximum values for your distance from Taipei 101.  Round to the nearest tenth

Smaller angle = Answer

Larger Angle = Answer

Answer 1

Without actually giving you the answer, I will give you the process for calculating the answer:

1. Recall the definition of Tangent with regard to a right triangle is Tan(θ)=opposite/adjacent
2. draw a triangle where the hight of the tower is the opposite side and the distance along the ground from you to the tower is the adjacent side.
3. With this drawing, you should be able to see that Tan(10)=height of tower/distance to tower
4. A little algebra will get you the expression: Distance to tower = Height of tower/Tan(10)
5. The original estimate for the angle of elevation was 10 degrees. Since this is only accurate to within 3 degrees it means that the actual angle is somewhere between 7 degrees and 13 degrees.
6. Use the expression from step (4) and substitute 7 for 10 to get the distance estimate for the smaller angle. Substitute 13 for 10 to get the distance estimate for the larger angle.

Best of luck to you. I hope these hints helped you arrive at the correct solutions.

Your answers for the smaller and larger angles should be about 2200 meters and 4100 meters. These are not exact answers, but rounded to the nearest hundred so that you can know if you are on the right track.

Remember to make sure your calculator is in Degree mode and not Radian mode.

Thank you,

David

Answer 2

Given that the angle of elevation is the angle between you and the top of the building:

Theta (angle of elevation) = 10°

y (height of building) = 509 m

y is the opposite angle of the right triangle formed by the building and you and the distance between.

x is the adjacent angle of the triangle.

tan(theta) = Opposite / adjacent

So,

tan(10°) = 509 / x

Solving for x,

509/ tan(10) = x

x = 2886.7 m

For smaller angle, it would be 7°

x = 4145.5 m

Larger angle, 13°

x = 2204.7

Hope this helps!

Answer 3

Hi Mike,

I would draw a right triangle with the vertical leg equal to 509 meters from the ground to the top of the building and the horizontal leg equal to x from the base of the building to where you are standing. The angle of elevation from where you are standing to the top of the building is 10 +/- 3 degrees. So, you have two equations to solve - one for 7 degrees and one for 13 degrees.

We know that the tangent of an angle in a right triangle = Side Opposite over Side Adjacent. So, the two equations are:

Tan 7deg = 509m/x

Tan 13deg = 509/x

Find the tangent of 7deg:

Tan 7deg = 0.871

Find the tangent of 13deg:

Tan 13deg = 0.463

Rearrange the equations:

Smaller angle (7deg): x = 509/0.871 = 584.4m

Larger angle (13deg): x = 509m/0.463 = 1,099.4m

I hope that helps!