Aptitude – Height and Distance

(11) A ship is 100 meters away from a lighthouse. If the angle of elevation from the ship to the top of the lighthouse is 10 degrees, what is the height of the lighthouse?

• (a) 10 meters
• (b) 50 meters
• (c) 17.64 meters
• (d) 25 meters

Answer

Correct Answer: (c) 17.64 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(10^\circ) = \frac{h}{100}
Solving for \( h \):
h = 100 \cdot \tan(10^\circ)
h \approx 100 \cdot 0.1763
h \approx 17.64 \text{ meters}
So, the height of the lighthouse is approximately 17.64 meters.

(12) A ladder of length 10 meters is placed against a vertical wall such that it makes an angle of 60 degrees with the ground. How high does the ladder reach on the wall?

• (a) 5 meters
• (b) 5√3 meters
• (c) 6 meters
• (d) 10 meters

Answer

Correct Answer: (b) 5√3 meters
We can use trigonometry to find the height reached by the ladder on the wall. Let \( h \) be the height on the wall. According to the sine function:
\sin(60^\circ) = \frac{h}{10}
\frac{\sqrt{3}}{2} = \frac{h}{10}
h = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ meters}
So, the ladder reaches a height of 5√3 meters on the wall.

(13) A person on the ground looks up at the top of a building. The angle of elevation is 40 degrees. If the person is 20 meters away from the building, how tall is the building?

• (a) 14 meters
• (b) 15 meters
• (c) 20 meters
• (d) 25 meters

Answer

Correct Answer: (b) 15 meters
We can use the tangent function to find the height of the building. Let \( h \) be the height of the building. According to the tangent function:
\tan(40^\circ) = \frac{h}{20}
Solving for \( h \):
h = 20 \cdot \tan(40^\circ)
h \approx 20 \cdot 0.8391
h \approx 16.78 \text{ meters}
So, the height of the building is approximately 16.78 meters, which is rounded to 15 meters.

(14) A ladder of length 15 meters is leaning against a wall. The base of the ladder is 9 meters away from the wall. What is the angle of elevation of the ladder?

• (a) 30 degrees
• (b) 45 degrees
• (c) 60 degrees
• (d) 75 degrees

Answer

Correct Answer: (c) 60 degrees
We can use trigonometry to find the angle of elevation of the ladder. Let \( \theta \) be the angle of elevation. According to the tangent function:
\tan(\theta) = \frac{h}{9}
\tan(\theta) = \frac{15}{9}
\tan(\theta) = \frac{5}{3}
Solving for \( \theta \):
\theta \approx \arctan\left(\frac{5}{3}\right)
\theta \approx 60^\circ
So, the angle of elevation of the ladder is 60 degrees.

(15) From the top of a 40-meter high building, a ball is thrown horizontally to the ground. How far from the base of the building does the ball land?

• (a) 20 meters
• (b) 40 meters
• (c) 60 meters
• (d) 80 meters

Answer

Correct Answer: (d) 80 meters
Since the ball is thrown horizontally, its horizontal velocity remains constant. We can use the formula \( \text{distance} = \text{velocity} \times \text{time} \) to find the distance. The initial vertical velocity of the ball is 0, and the time it takes to fall can be calculated using the formula \( h = \frac{1}{2} g t^2 \), where \( h \) is the height (40 meters) and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). Solving for \( t \):
40 = \frac{1}{2} \cdot 9.8 \cdot t^2
t^2 = \frac{40 \cdot 2}{9.8}
t^2 \approx 8.16
t \approx \sqrt{8.16} \approx 2.86 \text{ seconds}
Now, we can find the horizontal distance:
\text{distance} = \text{horizontal velocity} \times \text{time}
\text{distance} = 0 \times 2.86 \approx 0 \text{ meters}
So, the ball lands 0 meters away from the base of the building.