Height and Distance: Mastering Concepts for Competitive Exams

Height and Distance

Q. A 10-meter tall pole casts a shadow of 5 meters long. What is the angle of elevation of the sun?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

tan(angle) = height/shadow = 10/5 = 2, angle = 60 degrees.

Correct Answer: C — 60 degrees

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Q. A building casts a shadow of 10 meters when the angle of elevation of the sun is 30 degrees. What is the height of the building?

A. 5√3 meters
B. 10 meters
C. 15 meters
D. 20 meters

Solution

Height = shadow * tan(angle) = 10 * √3 = 5√3 meters.

Correct Answer: A — 5√3 meters

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Q. A building casts a shadow of 10 meters when the angle of elevation of the sun is 45 degrees. What is the height of the building?

A. 10 meters
B. 15 meters
C. 20 meters
D. 25 meters

Solution

Height = shadow * tan(45) = 10 * 1 = 10 meters.

Correct Answer: A — 10 meters

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Q. A building casts a shadow of 20 meters when the angle of elevation of the sun is 45 degrees. What is the height of the building?

A. 20 meters
B. 30 meters
C. 40 meters
D. 50 meters

Solution

Height = shadow * tan(45) = 20 * 1 = 20 meters

Correct Answer: A — 20 meters

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Q. A building is 120 meters tall. From a point 80 meters away from the base of the building, what is the angle of elevation to the top of the building?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

tan(θ) = height/distance = 120/80 = 1.5, θ = tan⁻¹(1.5) ≈ 60 degrees.

Correct Answer: C — 60 degrees

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Q. A building is 20 meters tall. If a person standing 15 meters away from the building sees the top at an angle of elevation of θ, what is sin(θ)?

A. 0.8
B. 0.6
C. 0.5
D. 0.75

Solution

sin(θ) = height / hypotenuse; hypotenuse = √(20^2 + 15^2) = √625 = 25; sin(θ) = 20/25 = 0.8

Correct Answer: A — 0.8

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Q. A building is 24 meters tall. If a person is standing 32 meters away from the base of the building, what is the angle of elevation to the top of the building?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

tan(θ) = height/distance = 24/32 = 0.75; θ = tan^(-1)(0.75) ≈ 36.87 degrees.

Correct Answer: C — 60 degrees

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Q. A building is 80 meters tall. From a point 60 meters away from the base of the building, the angle of elevation to the top of the building is what?

A. 45 degrees
B. 60 degrees
C. 30 degrees
D. 75 degrees

Solution

Using tan(θ) = height/distance, tan(θ) = 80/60, θ = tan⁻¹(4/3) ≈ 53.13 degrees.

Correct Answer: B — 60 degrees

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Q. A building is 80 meters tall. From a point 60 meters away from the base of the building, what is the angle of elevation to the top of the building?

A. 30 degrees
B. 36.87 degrees
C. 45 degrees
D. 53.13 degrees

Solution

tan(θ) = height/distance = 80/60, θ = tan⁻¹(4/3) ≈ 36.87 degrees.

Correct Answer: B — 36.87 degrees

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Q. A building is 80 meters tall. From a point on the ground, the angle of elevation to the top of the building is 60 degrees. How far is the point from the base of the building?

A. 40√3 meters
B. 80 meters
C. 60 meters
D. 100 meters

Solution

Distance = height / tan(angle) = 80 / √3 = 80√3/3 meters.

Correct Answer: A — 40√3 meters

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Q. A flagpole is 12 meters high. If a person standing 16 meters away from the base of the flagpole sees the top of the flagpole at an angle of elevation of θ, what is tan(θ)?

A. 0.75
B. 0.6
C. 0.5
D. 0.8

Solution

tan(θ) = height/base = 12/16 = 0.75.

Correct Answer: A — 0.75

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Q. A flagpole is 15 meters tall. From a point 10 meters away from the base of the flagpole, what is the angle of elevation to the top of the flagpole?

A. 30 degrees
B. 36.87 degrees
C. 45 degrees
D. 60 degrees

Solution

tan(θ) = height/distance = 15/10 => θ = tan⁻¹(1.5) = 56.31 degrees

Correct Answer: B — 36.87 degrees

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Q. A kite is flying at a height of 100 meters. If the angle of elevation from a point on the ground to the kite is 60 degrees, how far is the point from the base of the kite's height?

A. 50 meters
B. 100 meters
C. 150 meters
D. 200 meters

Solution

Distance = height / tan(angle) = 100 / √3 ≈ 50 meters.

Correct Answer: A — 50 meters

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Q. A kite is flying at a height of 100 meters. If the angle of elevation from a point on the ground to the kite is 60 degrees, how far is the point from the base of the kite?

A. 50 meters
B. 100 meters
C. 150 meters
D. 200 meters

Solution

Distance = height / tan(angle) = 100 / √3 = 100√3/3 meters.

Correct Answer: A — 50 meters

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Q. A kite is flying at a height of 30 meters. If the angle of elevation from a point on the ground to the kite is 60 degrees, how far is the point from the base of the kite?

A. 15√3 meters
B. 30 meters
C. 10√3 meters
D. 20 meters

Solution

Distance = height / tan(angle) = 30 / √3 = 10√3 meters.

Correct Answer: C — 10√3 meters

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Q. A kite is flying at a height of 30 meters. If the angle of elevation from a point on the ground to the kite is 60 degrees, how far is the point from the base of the kite's height?

A. 15√3 meters
B. 30 meters
C. 20 meters
D. 10√3 meters

Solution

Distance = height / tan(angle) = 30 / (√3/3) = 30√3/3 = 10√3 meters.

Correct Answer: A — 15√3 meters

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Q. A kite is flying at a height of 40 meters. If the string makes an angle of 60 degrees with the ground, how long is the string?

A. 20√3 meters
B. 40 meters
C. 80 meters
D. 60 meters

Solution

Length of string = height / sin(60) = 40 / (√3/2) = 80/√3 = 20√3 meters.

Correct Answer: A — 20√3 meters

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Q. A kite is flying at a height of 50 meters. If the angle of elevation from a point on the ground to the kite is 60 degrees, how far is the point from the base of the kite's height?

A. 25 meters
B. 30 meters
C. 35 meters
D. 40 meters

Solution

Distance = height / tan(angle) = 50 / √3 = 50 / 1.732 = 28.87 meters.

Correct Answer: A — 25 meters

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Q. A kite is flying at a height of 50 meters. If the angle of elevation from a point on the ground to the kite is 60 degrees, how far is the point from the base of the kite?

A. 25√3 meters
B. 50 meters
C. 50√3 meters
D. 75 meters

Solution

Distance = height / tan(angle) = 50 / (√3/3) = 50 * 3/√3 = 25√3 meters.

Correct Answer: A — 25√3 meters

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Q. A kite is flying at a height of 50 meters. If the angle of elevation from a point on the ground is 30 degrees, how far is the point from the base of the kite?

A. 50√3
B. 25√3
C. 75√3
D. 100√3

Solution

distance = height / tan(30) = 50 / (1/√3) = 50√3 meters

Correct Answer: A — 50√3

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Q. A kite is flying at a height of 60 meters. If the angle of elevation from a point on the ground to the kite is 45 degrees, how far is the point from the base of the kite?

A. 30 meters
B. 40 meters
C. 60 meters
D. 70 meters

Solution

Distance = height / tan(angle) = 60 / 1 = 60 meters.

Correct Answer: C — 60 meters

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Q. A kite is flying at a height of 60 meters. If the angle of elevation from a point on the ground to the kite is 30 degrees, how far is the point from the base of the kite?

A. 60√3 meters
B. 30√3 meters
C. 20√3 meters
D. 10√3 meters

Solution

Distance = height / tan(angle) = 60 / (1/√3) = 60√3 meters.

Correct Answer: A — 60√3 meters

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Q. A ladder 10 meters long leans against a wall. If the foot of the ladder is 6 meters away from the wall, how high does the ladder reach on the wall?

A. 8 meters
B. 9 meters
C. 10 meters
D. 7 meters

Solution

Using Pythagoras: height = √(10^2 - 6^2) = √(100 - 36) = √64 = 8 meters.

Correct Answer: A — 8 meters

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Q. A ladder 10 meters long reaches a window 8 meters above the ground. What is the angle of elevation of the ladder from the ground?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

Using sin(θ) = opposite/hypotenuse, sin(θ) = 8/10, θ = sin⁻¹(0.8) ≈ 53.13 degrees.

Correct Answer: C — 60 degrees

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Q. A ladder 10 meters long reaches a window 8 meters high. How far is the base of the ladder from the wall?

A. 6 meters
B. 4 meters
C. 5 meters
D. 3 meters

Solution

Using Pythagoras: base = √(10^2 - 8^2) = √(100 - 64) = √36 = 6 meters

Correct Answer: A — 6 meters

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Q. A ladder 10 meters long reaches a window 8 meters high. What is the angle of elevation of the ladder from the ground?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

Using sin(θ) = opposite/hypotenuse, sin(θ) = 8/10, θ = arcsin(0.8) = 53.13 degrees.

Correct Answer: C — 60 degrees

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Q. A ladder 20 meters long reaches a window 15 meters above the ground. What is the angle of elevation of the ladder from the ground?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

Using sin(θ) = opposite/hypotenuse, sin(θ) = 15/20, θ = sin⁻¹(0.75) ≈ 48.6 degrees.

Correct Answer: C — 60 degrees

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Q. A ladder 20 meters long reaches a window 16 meters above the ground. What is the angle of elevation of the ladder from the ground?

A. 30 degrees
B. 45 degrees
C. 60 degrees
D. 75 degrees

Solution

Using sin(θ) = opposite/hypotenuse, sin(θ) = 16/20, θ = sin⁻¹(0.8) ≈ 60 degrees.

Correct Answer: C — 60 degrees

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Q. A ladder 25 meters long leans against a wall. If the foot of the ladder is 7 meters away from the wall, how high does the ladder reach on the wall?

A. 24 meters
B. 20 meters
C. 21 meters
D. 22 meters

Solution

Using Pythagoras: height = √(25^2 - 7^2) = √(625 - 49) = √576 = 24 meters

Correct Answer: B — 20 meters

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Q. A ladder is leaning against a wall. If the foot of the ladder is 6 meters away from the wall and the top of the ladder reaches a height of 8 meters, what is the length of the ladder?

A. 10 meters
B. 12 meters
C. 14 meters
D. 16 meters

Solution

Using the Pythagorean theorem, length of ladder = √(6^2 + 8^2) = √(36 + 64) = √100 = 10 meters.

Correct Answer: A — 10 meters

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Height and Distance

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