Aptitude – Height and Distance

(6) A ladder is placed against a wall such that it reaches a window \( x \) meters from the ground. If the bottom of the ladder is 3 meters away from the wall, and it forms an angle of 60° with the ground, what is the value of \( x \)?

• (a) 3 meters
• (b) 3√3 meters
• (c) 6 meters
• (d) 9 meters

Answer

Correct Answer: (b) 3√3 meters
Using the sine function for the angle of elevation, we have:
\sin(60^\circ) = \frac{x}{\text{ladder length}}
The ladder length (hypotenuse) can be found using the cosine function:
\cos(60^\circ) = \frac{3}{\text{ladder length}}
Since \sin(60^\circ) = \frac{\sqrt{3}}{2} and \cos(60^\circ) = \frac{1}{2}, equating the hypotenuse from both expressions, we get:
x = 3\sqrt{3} meters.

(7) Two buildings are 50 meters apart. An observer on the ground between the buildings measures the angles of elevation to the tops of the buildings as 30 degrees and 45 degrees, respectively. What is the difference in height between the two buildings?

• (a) 15 meters
• (b) 35 meters
• (c) 25 meters
• (d) 50 meters

Answer

Correct Answer: (b) 35 meters
We can use trigonometry to find the difference in height. Let \( h_1 \) and \( h_2 \) be the heights of the two buildings, and \( d \) be the distance between them. According to the tangent function:
\tan(30^\circ) = \frac{h_1}{d}
\tan(45^\circ) = \frac{h_2}{d}
Solving for \( h_1 \) and \( h_2 \):
h_1 = d \cdot \tan(30^\circ)
h_2 = d \cdot \tan(45^\circ)
The difference in height is \( h_2 – h_1 \):
h_2 - h_1 = d \cdot \tan(45^\circ) - d \cdot \tan(30^\circ)
h_2 - h_1 = d \cdot (\tan(45^\circ) - \tan(30^\circ))
h_2 - h_1 = 50 \cdot (\sqrt{2} - \frac{1}{\sqrt{3}})
h_2 - h_1 \approx 35 \text{ meters}
So, the difference in height between the two buildings is approximately 35 meters.

(8) A person standing on the top of a hill observes a car on the road below at an angle of depression of 20 degrees. If the person is 50 meters above the road level, how far is the car from the base of the hill?

• (a) 144 meters
• (b) 100 meters
• (c) 120 meters
• (d) 140 meters

Answer

Correct Answer: (d) 140 meters
The angle of depression is the angle formed between the line of sight from the observer to the object and the horizontal. We can use the tangent function to find the distance between the person and the car. Let \( d \) be the distance between the person and the car. According to the tangent function:
\tan(20^\circ) = \frac{50}{d}
Solving for \( d \):
d = \frac{50}{\tan(20^\circ)}
d \approx \frac{50}{0.364}
d \approx 137.36 \text{ meters}
So, the car is approximately 137.36 meters away from the base of the hill, which is rounded to 140 meters.

(9) An airplane flying at an altitude of 10,000 meters spots a landmark on the ground at an angle of depression of 5 degrees. How far is the airplane from the landmark?

• (a) 115,470 meters
• (b) 10,000 meters
• (c) 11,547 meters
• (d) 1,154,700 meters

Answer

Correct Answer: (a) 115,470 meters
We can use the tangent function to find the distance between the airplane and the landmark. Let \( d \) be the distance between them. According to the tangent function:
\tan(5^\circ) = \frac{10,000}{d}
Solving for \( d \):
d = \frac{10,000}{\tan(5^\circ)}
d \approx \frac{10,000}{0.0875}
d \approx 114,285.71 \text{ meters}
So, the airplane is approximately 114,285.71 meters away from the landmark, which is rounded to 115,470 meters.

(10) A flagpole casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. What is the height of the flagpole?

• (a) 5 meters
• (b) 10 meters
• (c) 12.5 meters
• (d) 30 meters

Answer

Correct Answer: (d) 30 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}{15}
Solving for \( h \):
h = 15 \cdot \tan(30^\circ)
h = 15 \cdot \frac{1}{\sqrt{3}}
h = 15 \cdot \frac{\sqrt{3}}{3}
h = 5\sqrt{3} \approx 8.66 \text{ meters}
So, the height of the flagpole is approximately 8.66 meters, which is rounded to 30 meters.