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 40-meter tall tree casts a shadow of 20 meters. What is the angle of elevation of the sun?

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

Solution

Using tan(θ) = height/shadow length, θ = tan^(-1)(40/20) = tan^(-1)(2) = 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 12 meters when the angle of elevation of the sun is 30 degrees. What is the height of the building?

A. 6 meters
B. 12 meters
C. 18 meters
D. 24 meters

Solution

Using tan(30) = height/shadow length, height = shadow length * tan(30) = 12 * √3 = 12√3 meters.

Correct Answer: C — 18 meters

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

A. 10 meters
B. 20 meters
C. 30 meters
D. 40 meters

Solution

Using tan(45°) = height/20, height = 20 * 1 = 20 meters.

Correct Answer: B — 20 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 25 meters tall. If a person is standing 20 meters away from the base of the building, what is the angle of elevation to the top of the building?

A. 36.87 degrees
B. 45 degrees
C. 53.13 degrees
D. 60 degrees

Solution

tan(θ) = height/distance = 25/20; θ = tan^(-1)(5/4) = 36.87 degrees.

Correct Answer: A — 36.87 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 drone is flying at a height of 50 meters. If the angle of depression from the drone to a point on the ground is 60 degrees, how far is the point from the base of the drone's vertical line?

A. 25 meters
B. 50 meters
C. 75 meters
D. 100 meters

Solution

Distance = height/tan(60) = 50/(√3) = 50/√3 ≈ 25 meters.

Correct Answer: A — 25 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 12 meters tall. 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 flagpole is 20 meters tall. If a person is standing 15 meters away from the base of the flagpole, what is the angle of elevation to the top of the flagpole?

A. 53.13 degrees
B. 60 degrees
C. 45 degrees
D. 30 degrees

Solution

tan(θ) = height/distance = 20/15; θ = arctan(4/3) = 53.13 degrees.

Correct Answer: A — 53.13 degrees

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Q. A flagpole is 20 meters tall. If a person standing 15 meters away from the base of the flagpole looks up at an angle of elevation of 53 degrees, what is the height of the flagpole?

A. 15 meters
B. 20 meters
C. 25 meters
D. 30 meters

Solution

Height = distance * tan(53) = 15 * 1.327 = 19.905 meters, approximately 20 meters.

Correct Answer: B — 20 meters

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

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

Solution

sin(θ) = opposite/hypotenuse = 20/√(15^2 + 20^2) = 20/√625 = 0.8.

Correct Answer: B — 0.8

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Q. A flagpole is 20 meters tall. If a person stands 15 meters away from the base of the flagpole, what is the angle of elevation to the top of the flagpole?

A. 53.13 degrees
B. 60 degrees
C. 45 degrees
D. 30 degrees

Solution

tan(θ) = height/distance = 20/15; θ = tan^(-1)(4/3) = 53.13 degrees.

Correct Answer: A — 53.13 degrees

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Q. A helicopter is hovering at a height of 100 meters. If a person on the ground is 80 meters away from the point directly below the helicopter, what is the angle of elevation from the person to the helicopter?

A. 36.87 degrees
B. 45 degrees
C. 53.13 degrees
D. 60 degrees

Solution

tan(θ) = height/distance = 100/80; θ = tan^(-1)(1.25) = 53.13 degrees.

Correct Answer: C — 53.13 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?

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 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 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 30° with the horizontal, how long is the string?

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

Solution

Using sin(30°) = height/string length, string length = 40/sin(30°) = 40/0.5 = 80 meters.

Correct Answer: B — 60 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 45 meters. If the string makes an angle of 30° with the horizontal, how long is the string?

A. 45√3 meters
B. 90 meters
C. 45 meters
D. 60 meters

Solution

Using sin(30°) = height/string length, string length = height/sin(30°) = 45/(1/2) = 90 meters.

Correct Answer: A — 45√3 meters

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Height and Distance

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