Height and Distance: Mastering Concepts for Competitive Exams

Height and Distance

Q. A tower is 50 meters high. From a point 30 meters away from the base of the tower, what is the angle of elevation to the top of the tower?

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

Solution

tan(θ) = height / distance = 50 / 30; θ = tan^(-1)(5/3) ≈ 60 degrees

Correct Answer: C — 60 degrees

Q. A tower is 50 meters high. From a point 40 meters away from the base of the tower, what is the angle of elevation to the top of the tower?

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

Solution

tan(θ) = height/distance = 50/40, θ = tan⁻¹(1.25) ≈ 51.34 degrees.

Correct Answer: A — 36.87 degrees

Q. A tower is 50 meters high. From a point on the ground, the angle of elevation to the top of the tower is 45 degrees. How far is the point from the base of the tower?

A. 25 meters
B. 35 meters
C. 50 meters
D. 70 meters

Solution

Distance = height / tan(angle) = 50 / 1 = 50 meters.

Correct Answer: C — 50 meters

Q. A tree casts a shadow of 10 meters long. If the angle of elevation of the sun is 30 degrees, what is the height of the tree?

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

Solution

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

Correct Answer: A — 5√3 meters

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

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

Solution

Height = shadow * tan(angle) = 10 * √3 = 10 * 1.732 = 17.32 meters.

Correct Answer: C — 15 meters

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

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

Q. A tree is 15 meters tall. If the angle of elevation from a point 20 meters away from the base of the tree is θ, what is tan(θ)?

A. 0.75
B. 1.25
C. 1.5
D. 2

Solution

tan(θ) = height / distance = 15 / 20 = 0.75

Correct Answer: A — 0.75

Q. A tree is 30 meters tall. If the angle of elevation from a point 40 meters away from the base of the tree is θ, what is tan(θ)?

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

Solution

tan(θ) = height / distance = 30 / 40 = 0.75

Correct Answer: A — 0.75

Q. From a point 40 meters away from the base of a tower, the angle of elevation to the top of the tower is 30 degrees. What is the height of the tower?

A. 20√3 meters
B. 30 meters
C. 40 meters
D. 50 meters

Solution

Height = distance * tan(30) = 40 * (1/√3) = 40/√3 = 20√3 meters

Correct Answer: A — 20√3 meters

Q. From a point 60 meters away from the base of a hill, the angle of elevation to the top of the hill is 30 degrees. What is the height of the hill?

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

Solution

Height = distance * tan(angle) = 60 * (1/√3) = 60 / 1.732 = 34.64 meters.

Correct Answer: B — 30 meters

Q. From a point 80 meters away from the base of a tower, the angle of elevation to the top of the tower is 30 degrees. What is the height of the tower?

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

Solution

Height = distance * tan(angle) = 80 * (1/√3) = 80 / 1.732 = 46.19 meters.

Correct Answer: C — 40 meters

Q. From a point on the ground, the angle of elevation to the top of a 12-meter tall pole is 60 degrees. How far is the point from the base of the pole?

A. 6 meters
B. 10 meters
C. 12 meters
D. 15 meters

Solution

Distance = height / tan(60) = 12 / √3 ≈ 6 meters.

Correct Answer: A — 6 meters

Q. From a point on the ground, the angle of elevation to the top of a 20-meter tall building is 45 degrees. How far is the point from the base of the building?

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

Solution

Distance = height / tan(angle) = 20 / 1 = 20 meters.

Correct Answer: B — 20 meters

Q. From a point on the ground, the angle of elevation to the top of a 40-meter tall building is 60 degrees. How far is the point from the base of the building?

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

Solution

Distance = height / tan(angle) = 40 / √3 = 40 / 1.732 = 23.09 meters.

Correct Answer: A — 20 meters

Q. From a point on the ground, the angle of elevation to the top of a 50-meter tall building is 45 degrees. How far is the point from the base of the building?

A. 50 meters
B. 25 meters
C. 35 meters
D. 40 meters

Solution

distance = height / tan(45) = 50 / 1 = 50 meters

Correct Answer: A — 50 meters

Q. From a point on the ground, the angle of elevation to the top of a building is 45 degrees. If the point is 20 meters away from the building, what is the height of the building?

A. 20 meters
B. 10 meters
C. 30 meters
D. 25 meters

Solution

Height = distance * tan(45) = 20 * 1 = 20 meters.

Correct Answer: A — 20 meters

Q. From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the hill is 100 meters high, how far is the point from the base of the hill?

A. 100√3 meters
B. 200 meters
C. 100 meters
D. 50 meters

Solution

Distance = height / tan(angle) = 100 / (1/√3) = 100√3 meters.

Correct Answer: A — 100√3 meters

Q. From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the distance from the point to the base of the hill is 50 meters, what is the height of the hill?

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

Solution

Height = distance * tan(30) = 50 * (1/√3) ≈ 28.87 meters.

Correct Answer: A — 25 meters

Q. From a point on the ground, the angle of elevation to the top of a hill is 30 degrees. If the point is 100 meters away from the base of the hill, what is the height of the hill?

A. 50 meters
B. 100 meters
C. 100√3 meters
D. 50√3 meters

Solution

Height = distance * tan(angle) = 100 * (1/√3) = 100/√3 = 50√3 meters.

Correct Answer: D — 50√3 meters

Q. From a point on the ground, the angle of elevation to the top of a hill is 37 degrees. If the point is 50 meters away from the base of the hill, what is the height of the hill?

A. 30 meters
B. 37 meters
C. 40 meters
D. 50 meters

Solution

Height = distance * tan(angle) = 50 * tan(37°) ≈ 50 * 0.753 = 37.65 meters.

Correct Answer: B — 37 meters

Q. From a point on the ground, the angle of elevation to the top of a hill is 37 degrees. If the distance from the point to the base of the hill is 50 meters, what is the height of the hill?

A. 30 meters
B. 40 meters
C. 37 meters
D. 25 meters

Solution

Height = distance * tan(angle) = 50 * tan(37) ≈ 30 meters.

Correct Answer: A — 30 meters

Q. From a point on the ground, the angle of elevation to the top of a tower is 45 degrees. If the point is 10 meters away from the base of the tower, what is the height of the tower?

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

Solution

Height = distance * tan(45) = 10 * 1 = 10 meters

Correct Answer: A — 10 meters

Q. From a point on the ground, the angle of elevation to the top of a tree is 45 degrees. If the tree is 10 meters tall, how far is the point from the base of the tree?

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

Solution

Distance = height / tan(angle) = 10 / 1 = 10 meters.

Correct Answer: A — 10 meters

Q. From a point on the ground, the angle of elevation to the top of a tree is 45 degrees. If the point is 20 meters away from the base of the tree, what is the height of the tree?

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

Solution

Height = distance * tan(angle) = 20 * 1 = 20 meters.

Correct Answer: A — 20 meters

Q. From the top of a 100-meter high cliff, the angle of depression to a boat in the sea is 30 degrees. How far is the boat from the base of the cliff?

A. 100 meters
B. 150 meters
C. 173 meters
D. 200 meters

Solution

Distance = height / tan(angle) = 100 / (1/√3) = 100 * √3 = 173.21 meters.

Correct Answer: C — 173 meters

Q. From the top of a 40-meter high tower, the angle of depression to a point on the ground is 60 degrees. How far is the point from the base of the tower?

A. 20√3 meters
B. 40 meters
C. 30 meters
D. 50 meters

Solution

Distance = height / tan(angle) = 40 / (1/√3) = 40√3 meters.

Correct Answer: A — 20√3 meters

Q. From the top of a 40-meter high tower, the angle of depression to a point on the ground is 45 degrees. How far is the point from the base of the tower?

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

Solution

distance = height / tan(45) = 40 / 1 = 40 meters

Correct Answer: A — 40 meters

Q. From the top of a 50-meter high building, the angle of depression to a point on the ground is 30 degrees. How far is the point from the base of the building?

A. 25√3 meters
B. 50 meters
C. 75 meters
D. 100 meters

Solution

Distance = height / tan(angle) = 50 / (1/√3) = 50√3 meters.

Correct Answer: A — 25√3 meters

Q. From the top of a hill, the angle of depression to a car parked on the ground is 45 degrees. If the height of the hill is 100 meters, how far is the car from the base of the hill?

A. 100 meters
B. 150 meters
C. 200 meters
D. 250 meters

Solution

Distance = height / tan(angle) = 100 / 1 = 100 meters.

Correct Answer: A — 100 meters

Q. From the top of a hill, the angle of depression to a car parked on the ground is 60 degrees. If the height of the hill is 80 meters, how far is the car from the base of the hill?

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

Solution

Distance = height / tan(angle) = 80 / (√3/3) = 80√3/3 = 40√3 meters.

Correct Answer: A — 40√3 meters

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