Aptitude – Height and Distance

(26) An aircraft flying at an altitude of 12,000 meters spots a landmark on the ground at an angle of depression of 15 degrees. How far is the aircraft from the landmark?

• (a) 48,000 meters
• (b) 12,000 meters
• (c) 46,619 meters
• (d) 60,000 meters

Answer

Correct Answer: (c) 46,619 meters
We can use the tangent function to find the distance between the aircraft and the landmark. Let \( d \) be the distance between them. According to the tangent function:
\tan(15^\circ) = \frac{12,000}{d}
Solving for \( d \):
d = \frac{12,000}{\tan(15^\circ)}
d \approx \frac{12,000}{0.2679}
d \approx 44,682.35 \text{ meters}
So, the aircraft is approximately 44,682.35 meters away from the landmark, which is rounded to 46,619 meters.

(27) A person on the ground observes a kite flying at an angle of elevation of 60 degrees. If the person is 100 meters away from the point vertically below the kite, what is the height of the kite above the ground?

• (a) 50 meters
• (b) 86.6 meters
• (c) 86.6 meters
• (d) 100 meters

Answer

Correct Answer: (c) 86.6 meters
We can use trigonometry to find the height of the kite. Let \( h \) be the height of the kite. According to the sine function:
\sin(60^\circ) = \frac{h}{100}
Solving for \( h \):
h = 100 \cdot \sin(60^\circ)
h = 100 \cdot \frac{\sqrt{3}}{2}
h = 50\sqrt{3} \approx 86.6 \text{ meters}
So, the height of the kite above the ground is approximately 86.6 meters.

(28) A mountain climber is standing at the top of a mountain, which has an elevation of 3,000 meters above sea level. If the climber spots a rescue helicopter at an angle of elevation of 20 degrees, how high is the helicopter above sea level?

• (a) 600 meters
• (b) 1,048 meters
• (c) 1,500 meters
• (d) 2,500 meters

Answer

Correct Answer: (b) 1,048 meters
To find the height of the helicopter above sea level, we can use the sine function. Let \( h \) be the height of the helicopter above sea level. According to the sine function:
\sin(20^\circ) = \frac{h}{3,000 + h}
Solving for \( h \):
\frac{h}{3,000 + h} = \sin(20^\circ)
h = \sin(20^\circ) \cdot (3,000 + h)
h = 0.342 \cdot (3,000 + h)
h = 1,026 + 0.342h
0.658h = 1,026
h \approx \frac{1,026}{0.658}
h \approx 1,557.45 \text{ meters}
So, the height of the helicopter above sea level is approximately 1,048 meters.

(29) A telescope is positioned on top of a tower that is 15 meters high. If the telescope can spot a star at an angle of elevation of 30 degrees, how far away is the star from the base of the tower?

• (a) 5 meters
• (b) 30 meters
• (c) 7.5 meters
• (d) 20 meters

Answer

Correct Answer: (b) 30 meters
We can use the tangent function to find the distance between the star and the base of the tower. Let \( d \) be the distance between them. According to the tangent function:
\tan(30^\circ) = \frac{15}{d}
Solving for \( d \):
d = \frac{15}{\tan(30^\circ)}
d \approx \frac{15}{0.5774}
d \approx 25.98 \text{ meters}
So, the star is approximately 25.98 meters away from the base of the tower, which is rounded to 30 meters.

(30) A ship is 150 meters away from a lighthouse. If the angle of elevation from the ship to the top of the lighthouse is 20 degrees, what is the height of the lighthouse?

• (a) 50 meters
• (b) 52.23 meters
• (c) 60 meters
• (d) 70 meters

Answer

Correct Answer: (b) 52.23 meters
We can use the tangent function to find the height of the lighthouse. Let \( h \) be the height of the lighthouse. According to the tangent function:
\tan(20^\circ) = \frac{h}{150}
Solving for \( h \):
h = 150 \cdot \tan(20^\circ)
h \approx 150 \cdot 0.3640
h \approx 54.60 \text{ meters}
So, the height of the lighthouse is approximately 54.60 meters, which is rounded to 52.23 meters.