Heights and Distances Applications

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Q1. A building is 20 meters tall. If the angle of elevation from a point 10 meters away from the base of the building is θ, what is tan(θ)?

Solution:

tan(θ) = height/distance = 20/10 = 2.

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Q2. A person is standing 12 meters away from a vertical pole. If the angle of elevation to the top of the pole is 45 degrees, what is the height of the pole?

Solution:

Using tan(45) = height / 12, height = 12 * tan(45) = 12 * 1 = 12 meters.

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Q3. A tree casts a shadow of 15 meters when the angle of elevation of the sun is 30 degrees. How tall is the tree?

Solution:

Using the tangent function, tan(30) = height / 15. Therefore, height = 15 * tan(30) = 15 * (1/√3) = 5√3 meters.

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Q4. A 12-meter tall building casts a shadow of 8 meters. What is the angle of elevation of the sun?

Solution:

Using tan(θ) = height/shadow = 12/8 = 1.5, θ = tan⁻¹(1.5) ≈ 36.87 degrees.

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Q5. From a point 25 meters away from the base of a building, the angle of elevation to the top of the building is 30 degrees. What is the height of the building?

Solution:

Using tan(30) = height / 25, height = 25 * tan(30) = 25 * (1/√3) = 25/√3 meters.

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Q6. From a point on the ground, the angle of elevation to the top of a hill is 45 degrees. If the point is 50 meters away from the base of the hill, what is the height of the hill?

Solution:

Using tan(45°) = height/50, height = 50 * tan(45°) = 50 meters.

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Q7. A ladder 10 meters long leans against a wall. If the foot of the ladder is 6 meters away from the wall, what is the height at which the ladder touches the wall?

Solution:

Using Pythagorean theorem: height = √(10² - 6²) = √(100 - 36) = √64 = 8 meters.

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Q8. A building is 20 meters tall. If the angle of elevation from a point on the ground 10 meters away from the base of the building is θ, what is tan(θ)?

Solution:

tan(θ) = opposite/adjacent = 20/10 = 2.

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Q9. A ladder 25 meters long leans against a wall. If the foot of the ladder is 7 meters from the wall, what is the angle of elevation of the ladder?

Solution:

Using cos(θ) = adjacent/hypotenuse, cos(θ) = 7/25. Therefore, θ = cos⁻¹(7/25) which is approximately 60 degrees.

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Q10. If the angle of elevation to the top of a tower from a point 40 meters away is 30 degrees, what is the height of the tower?

Solution:

Using tan(30°) = height/40, height = 40 * (1/√3) = 40/√3 = 20√3 meters.

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