Trigonometry Real-World Applications 4

Question 1

A surveyor measures a plot of triangular land with two sides of 200 m and 150 m enclosing an angle of 72°. What is the area of the plot?

Accepted Answer

14,266 m²

Answer

Area = ½ × 200 × 150 × sin(72°) ≈ ½ × 200 × 150 × 0.9511 ≈ 14,266 m².

Question 2

A communications tower casts a 240-foot shadow when the sun's elevation angle is 32°. How tall is the tower?

Accepted Answer

150 feet

Answer

Tower height = 240 × tan(32°) ≈ 240 × 0.6249 ≈ 150 feet.

Question 3

A boat travels 15 km on a bearing of 040° and then 20 km on a bearing of 130°. How far is the boat from its starting point?

Accepted Answer

25 km

Answer

The angle between the two legs is 130° − 40° = 90°, so by Pythagorean theorem: √(15² + 20²) = √(225 + 400) = √625 = 25 km.

Question 4

An engineer models tidal water depth as d(t) = 4 + 3 sin(π t / 6) meters, where t is hours after midnight. What is the water depth at t = 3 hours?

Accepted Answer

7 meters

Answer

d(3) = 4 + 3 sin(π × 3 / 6) = 4 + 3 sin(π/2) = 4 + 3 × 1 = 7 meters.

Question 5

A zip line starts from a platform 50 feet high and ends at ground level 200 feet away horizontally. What angle does the zip line make with the horizontal?

Accepted Answer

14.0°

Answer

Angle = arctan(50/200) = arctan(0.25) ≈ 14.0°.

Trigonometry Practice Test

Trigonometry Practice Test

Trigonometry Real-World Applications 4