A pilot flying at an altitude of 3.7 km sights 2 control towers directly in front of him.

Question

A pilot flying at an altitude of 3.7 km sights 2 control towers directly in front of him. The angle of depression to the base of one tower is 55°, and the angle of depression to the base of the other tower is 35°. a) How far is the pilot from the first tower? Round to the nearest hundredth. b) How far is the pilot from the second tower? Round to the nearest hundredth. c) What is the distance between the 2 towers? Round to the nearest hundredth. d) If the pilot is traveling at a rate of 100 m/s, then how long does it take the pilot to fly over each tower?

Answer 1

(a) Angle of depression is always the angle from the horizontal looking down. If you create a right triangle, one leg will be 3.7 (altitude) and you are trying to find the hypotenuse (length from pilot to tower #1). This will involve sine, cosine, or tangent. Sine = opposite/hypotenuse; Cosine = adjacent/hypotenuse; Tangent = opposite/adjacent. Since the reference angle inside the right triangle is 35 degrees (90 - 55), then 3.7 is adjacent and x is the hypotenuse. So you would use cosine in this case. Cos(35) = 3.7/x. Solve for x. So x = 3.7/cos(35). Plug this into your calculator and you get 4.51 km. (*make sure the mode is in "degrees").

(b) Same thought process as (a). But now the angle of depression is 35 degrees, which makes the angle inside the right triangle 55 degrees (90 - 35). So Cos(55) = 3.7/y, where y is the length from the pilot to tower #2. So, y = 3.7/cos(55). Plug this into your calculator and you get 6.45 km.

(c) To find the length between the towers, you would want to find the whole length opposite of the pilot for both the big right triangle and the small right triangle and subtract those. Since you want to find the opposite and you have the adjacent, using tangent. So tan(35) = a/3.7. So a = 3.7*tan(35) --> a = 2.59077

To find the other opposite side use tangent again. Tan (55) = b/3.7. So b = 3.7*tan(55) --> b = 5.28415

Now subtract 5.28415-2.59077 = 2.69 km. So the towers are 2.69 km apart.

(d) Change 100 m/s into km/s since all the distances are in km. There are 1,000 meters in a kilometer. So 100/1000 = 0.1 km/s.

Since the horizontal length from the pilot to tower #1 is 2.59077 km, then 2.59077 ÷ 0.1 = 25.91 seconds.

To tower #2: 5.28415 ÷ 0.1 = 52.84 seconds

I hope this helps!

Answer 2

The video is a little messy because I was writing with a tablet, so I'll write the math here,

a. 90-55=35

SOH CAH TOA ---- Tan(angle)= opposite/adjacent

tan (35) = a/3.7

a= 2.59km

b. 90-35=55

Tan (35) = b/3.7

b= 5.28km

c. 5.28-2.59 = 2.69 km

d. turn 2.59 km into meters = 2590 meters divide by 100 m/s to get seconds to the first tower

2590/100 = 25.9 seconds

turn 5.28 km into meters = 5280 meters. divide by 100 m/s to get seconds to second tower

5280/100 = 52.5 seconds