Statue Height Quiz: Statue Height From Two Angles

1) With d and θ_t fixed, increasing θ_b increases H.

True

False

H = d(tanθ_b − tanθ_t) increases with tanθ_b.

Explanation

H = d(tanθ_b − tanθ_t) increases with tanθ_b.

2) Select the mistakes to avoid.

Using tan(θ_b − θ_t) instead of tanθ_b − tanθ_t

Assuming θ_b < θ_t when observer is above the top

Ignoring that tan relates vertical drop to horizontal distance

Assuming θ_t = 0° always implies H = h without checking geometry

Treating angle of elevation and depression as unequal along one sight line

Height uses difference of tangents; θ_b>θ_t; tan gives vertical/horizontal; θ_t=0° implies top at eye level only if geometry matches; depression equals elevation across a line.

Explanation

Height uses difference of tangents; θ_b>θ_t; tan gives vertical/horizontal; θ_t=0° implies top at eye level only if geometry matches; depression equals elevation across a line.

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4) Angles of depression are 33° (base) and 7° (top); d = 42 m. Find H.

22.12 m

25.12 m

19.12 m

28.12 m

H = 42( tan33° − tan7° ) = 22.12 m.

Explanation

H = 42( tan33° − tan7° ) = 22.12 m.

5) An observer is 20 m above ground with θ_b = 35° and θ_t = 18°. The statue’s height is ____ m.

d = h/tan35° = 28.56 m, then H = 20 − d·tan18° = 10.72 m.

Explanation

d = h/tan35° = 28.56 m, then H = 20 − d·tan18° = 10.72 m.

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6) Which statements are always true here?

θ_b > θ_t when observer is above the statue’s top

If θ_b = θ_t then H = 0

If θ_t = 0°, H = h automatically

If d doubles with angles fixed, H doubles

If angles fixed, h is determined by d and θ_b

θ_b>θ_t in this geometry; equal angles give zero height; H scales with d; with fixed angles, h = d·tanθ_b so h is not independent.

Explanation

θ_b>θ_t in this geometry; equal angles give zero height; H scales with d; with fixed angles, h = d·tanθ_b so h is not independent.

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7) Angles of depression are measured from the horizontal downwards but are treated as positive acute angles in tan-relations.

True

False

We use right-triangle ratios with positive acute angles for tangent.

Explanation

We use right-triangle ratios with positive acute angles for tangent.

8) A statue is 18 m tall. Angles of depression are 30° to base and 8° to top. Find d, the horizontal distance.

41.21 m

36.21 m

46.21 m

51.21 m

d = H/(tan30° − tan8°) = 41.21 m by rearranging H = d(tanθ_b − tanθ_t).

Explanation

d = H/(tan30° − tan8°) = 41.21 m by rearranging H = d(tanθ_b − tanθ_t).

9) Find θ_b if h = 24 m and d = 55 m (nearest degree).

24°

26°

22°

29°

tanθ_b = h/d = 24/55 ⇒ θ_b = arctan(24/55) ≈ 23.57°.

Explanation

tanθ_b = h/d = 24/55 ⇒ θ_b = arctan(24/55) ≈ 23.57°.

10) If h = 26 m, d = 40 m, and H = 18 m, find θ_t (nearest degree).

11°

14°

8°

17°

tanθ_t = (h − H)/d = (26−18)/40 = 0.2 ⇒ θ_t ≈ arctan(0.2) ≈ 11.31°.

Explanation

tanθ_t = (h − H)/d = (26−18)/40 = 0.2 ⇒ θ_t ≈ arctan(0.2) ≈ 11.31°.

11) Knowing only h and θ_b is sufficient to find H.

True

False

You also need θ_t or d; otherwise H cannot be determined.

Explanation

You also need θ_t or d; otherwise H cannot be determined.

12) From a rooftop, the angles of depression to the base and the top of a statue are 35° and 12°. The horizontal distance to the statue is 60 m. What is the statue’s height?

29.26 m

34.26 m

24.26 m

39.26 m

H = d(tan35°−tan12°) = 60( tan35° − tan12° ) = 29.26 m. Vertical drop to base minus drop to top equals statue height.

Explanation

H = d(tan35°−tan12°) = 60( tan35° − tan12° ) = 29.26 m. Vertical drop to base minus drop to top equals statue height.

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14) Which sets of givens suffice to determine H?

(d, θ_b, θ_t)

(h, θ_b, θ_t)

(h, θ_b)

(d, θ_t)

(h, d)

Either knowing d with both angles or knowing h with both angles is sufficient to compute H uniquely.

Explanation

Either knowing d with both angles or knowing h with both angles is sufficient to compute H uniquely.

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15) If both θ_b and θ_t double while d stays fixed, then H doubles.

True

False

Tangent is nonlinear; doubling angles does not double their tangents, so H does not simply double.

Explanation

Tangent is nonlinear; doubling angles does not double their tangents, so H does not simply double.

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17) The angle of depression equals the corresponding angle of elevation along the same line of sight.

True

False

By alternate interior angles with parallel horizontals, depression equals elevation.

Explanation

By alternate interior angles with parallel horizontals, depression equals elevation.

18) Select all correct height formulas for angles θ_b (base) and θ_t (top).

H = d( tan θ_b − tan θ_t )

H = h − d·tan θ_t

If h unknown, H = d( tan θ_b − tan θ_t )

H = h·(tan θ_t / tan θ_b)

H = h − d·tan θ_b

Drop to base is d·tanθ_b; to top is d·tanθ_t; their difference is H. If h is known, H=h−d·tanθ_t.

Explanation

Drop to base is d·tanθ_b; to top is d·tanθ_t; their difference is H. If h is known, H=h−d·tanθ_t.

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19) Observer’s eye is 30 m above ground. Angles of depression are 40° (base) and 10° (top). The statue’s height is ____ m (2 d.p.).

d=30/tan40°. Then H=30−d·tan10° = 30(1 − tan10°/tan40°) = 23.70 m.

Explanation

d=30/tan40°. Then H=30−d·tan10° = 30(1 − tan10°/tan40°) = 23.70 m.

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20) Given h, θ_b, θ_t, which workflows are valid?

Compute d = h/tanθ_b, then H = h − d·tanθ_t

Directly use H = h·(tanθ_b − tanθ_t)

Use H = d( tanθ_b − tanθ_t ) with d = h/tanθ_b

Set H = d·tanθ_t − h

Use d = h·tanθ_b

From tanθ_b=h/d ⇒ d = h/tanθ_b; then either compute H via h − d·tanθ_t or substitute d into d(tanθ_b−tanθ_t).

Explanation

From tanθ_b=h/d ⇒ d = h/tanθ_b; then either compute H via h − d·tanθ_t or substitute d into d(tanθ_b−tanθ_t).

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