Applications involving Right Triangles

A surveyor is standing 115 ft from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3 degrees. What is the height of the Washington Monument? Photo credit: Retrieved address.jpg&imgr efurl= com/memorial /washington -monument- address

Draw a diagram. What are you looking for? What sides are of interest ? What trigonometric function can you use? 115 ft 78.3 h o a

We want to find the opposite side and we know the adjacent side. Therefore we can use the tangent function. Tan =

So, o =( a ) ( tan 78.3 degrees) = (115 ft) ( ) = 555 ft Answer: The Washington Monument is 555 ft tall.

Skateboard problem A skateboard ramp has an angle of elevation of 18.4 degrees. The vertical side is 4 ft high. What is the length of the skateboard ramp? First draw a diagram.

Diagram 4 ft 18.4 degrees h a o What sides interest us? What trigonometric function can we use?

We know the length of the opposite side and we want to find the length of the hypotenuse. So we can use the sin function Sin 18.4 = h = = = 12.7 ft The length of the ramp is 12.7 ft

Homework problem: # 64 p 310 A 6 ft person walks from the base of a tower directly toward the tip of the shadow cast by the tower. When the person is 132 ft from the tower and 3 ft from the tip of the shadow, the person’s shadow starts to appear beyond the tower’s shadow. What is the height of the tower? a) Draw a DIAGRAM Photo credit:”moshilove photostream”

Diagram: What are the sides that interest us? 132 ft 6 ft 3 ft h a o A B C E D

b) Use a trigonometric function to write an equation involving the unknown quantity?

Tan = = = 2 In triangle EDC

In triangle ABC tan = In triangle EDC, tan = 2, so = 2, so o = 2 (135 ft) = 270 ft

The height of the tower is 270 ft. Note: The height of the Eiffel Tower is 1,050 ft 0r 520 m. Find the height of the tower without using a trigonometric function.

By what postulate are triangle ABC and triangle EDC similar? What do you you conclude about their sides? E B D C B D C A

Angle B and D are right angles, so they are congruent and angle C angle C (reflexive property of congruence); so the triangles ABC and EDC are similar by the AA Postulate. Therefore, the corresponding sides of the triangles are proportional. =, so o = = 270 ft

Homework problem: # 66, p. 310 A biologist wants to know the width w of a river so in order to properly set instruments for studying the pollutants in the water. From point A, the biologist walks downstream 100 ft and sights to point C. From this sighting, it is determined that angle theta equals 54 degrees. How wide is the river?

Diagram w A100 ft river C

tan 54 degrees = w/ 100ft so w = (tan 54 degrees )(100) = ( )(100) = ( )(100) = ft = ft The river is ft wide.

Homework Problem: # 67 p. 310 Photo credit: kawwsu19’s photostream One end of a steel cable zip-line is attached to a platform on top of a 150 ft pole. The other end of the zip-line is attached to the top of a 5-ft stake. The angle of elevation of the platform is 23 degrees. a) How long is the zip-line?

b )How far is the stake from the pole? c) Contestants take an avarege of 6 seconds to reach the ground from the top of the zip-line. At what rate are contestants moving down the line? At what rate are they dropping vertically?