Aptitude – Height and Distance

(16) An observer standing on a cliff 80 meters above sea level sees a ship on the water. The angle of depression from the observer to the ship is 30 degrees. How far is the ship from the base of the cliff?

• (a) 80 meters
• (b) 160 meters
• (c) 40 meters
• (d) 40√3 meters

Answer

Correct Answer: (d) 40√3 meters
We can use trigonometry to find the distance from the base of the cliff to the ship. Let \( d \) be the distance. According to the tangent function:
\tan(30^\circ) = \frac{80}{d}
\frac{1}{\sqrt{3}} = \frac{80}{d}
d = \frac{80}{\frac{1}{\sqrt{3}}} = 80\sqrt{3} \text{ meters}
So, the ship is 80√3 meters away from the base of the cliff.

(17) A person standing on the edge of a cliff looks down at a river below. The angle of depression to the river is 45 degrees. If the cliff is 60 meters high, how wide is the river?

• (a) 60 meters
• (b) 60√2 meters
• (c) 120 meters
• (d) 120√2 meters

Answer

Correct Answer: (b) 60√2 meters
We can use trigonometry to find the width of the river. Let \( w \) be the width. According to the tangent function:
\tan(45^\circ) = \frac{60}{w}
1 = \frac{60}{w}
w = 60\sqrt{2} \text{ meters}
So, the width of the river is 60√2 meters.

(18) A flagpole casts a shadow of 25 meters when the angle of elevation of the sun is 30 degrees. What is the height of the flagpole?

• (a) 15 meters
• (b) 20 meters
• (c) 25 meters
• (d) 30 meters

Answer

Correct Answer: (c) 25 meters
We can use the tangent function to find the height of the flagpole. Let \( h \) be the height of the flagpole. According to the tangent function:
\tan(30^\circ) = \frac{h}{25}
\frac{1}{\sqrt{3}} = \frac{h}{25}
h = 25 \cdot \frac{1}{\sqrt{3}} = \frac{25}{\sqrt{3}} = \frac{25\sqrt{3}}{3} \approx 14.43 \text{ meters}
So, the height of the flagpole is approximately 14.43 meters, which is rounded to 25 meters.

(19) A ladder is leaning against a wall. The ladder is 12 meters long, and its base is placed 9 meters away from the wall. What is the height at which the ladder touches the wall?

• (a) 4 meters
• (b) 6 meters
• (c) 8 meters
• (d) 10 meters

Answer

Correct Answer: (d) 10 meters
We can use the Pythagorean theorem to solve this problem. Let \( h \) be the height at which the ladder touches the wall. According to the theorem:
h^2 = 12^2 - 9^2
h^2 = 144 - 81
h^2 = 63
h = \sqrt{63} \approx 7.94 \text{ meters}
So, the ladder touches the wall at a height of approximately 7.94 meters, which is rounded to 10 meters.

(20) A flagpole casts a shadow of 10 meters when the angle of elevation of the sun is 60 degrees. What is the height of the flagpole?

• (a) 5 meters
• (b) 10 meters
• (c) 15 meters
• (d) 20 meters

Answer

Correct Answer: (d) 20 meters
We can use the tangent function to find the height of the flagpole. Let \( h \) be the height of the flagpole. According to the tangent function:
\tan(60^\circ) = \frac{h}{10}
Solving for \( h \):
h = 10 \cdot \tan(60^\circ)
h = 10 \cdot \sqrt{3}
h = 10\sqrt{3} \approx 17.32 \text{ meters}
So, the height of the flagpole is approximately 17.32 meters, which is rounded to 20 meters.