A surveyor stands at point A on horizontal ground and measures the angle of elevation to t… — ACT

A surveyor stands at point $A$ on horizontal ground and measures the angle of elevation to the top of a vertical cliff, point $C$ , such that $\tan(\angle CAD) = \frac{1}{2}$ , where $D$ is the base of the cliff directly below $C$ . The surveyor then walks $50$ feet closer to the cliff along a straight horizontal path to point $B$ , where the angle of elevation to point $C$ satisfies $\tan(\angle CBD) = \frac{4}{3}$ . Points $A$ , $B$ , and $D$ are collinear. What is the height, in feet, of the cliff?

A
25
B
30
40Answer
D
60
E
75

Answer

40 feet

By applying SOHCAHTOA to both right triangles, we set up

\tan(\angle CBD) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{BD} = \frac{4}{3}

, giving

BD = \frac{3}{4}h

. For the larger triangle,

\tan(\angle CAD) = \frac{h}{50 + BD} = \frac{1}{2}

. Substituting

BD = \frac{3}{4}h

into the equation gives

50 + \frac{3}{4}h = 2h

, which solves directly to

h = 40

feet.

Step-by-Step Solution

Define the unknown quantities using the right triangles formed by the cliff and the observation points.

Let

h = CD

be the height of the cliff, and let

d = BD

be the horizontal distance from point

B

to the cliff base

D

. The total distance from

A

D

AD = AB + BD = 50 + d

Establishing explicit variables allows us to translate the geometric relationships into algebraic equations.

Apply the tangent ratio (SOHCAHTOA:

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

) to right triangle

BCD

\tan(\angle CBD) = \frac{CD}{BD} \implies \frac{4}{3} = \frac{h}{d} \implies d = \frac{3}{4}h

Expressing the distance

d

in terms of height

h

enables substitution into the second right triangle equation.

Apply the tangent ratio to right triangle

ACD

and substitute

d = \frac{3}{4}h

\tan(\angle CAD) = \frac{CD}{AD} \implies \frac{1}{2} = \frac{h}{50 + d} \implies \frac{1}{2} = \frac{h}{50 + \frac{3}{4}h}

This creates a single linear equation in terms of the cliff height

h

Solve the equation for

h

50 + \frac{3}{4}h = 2h \implies 50 = 2h - \frac{3}{4}h \implies 50 = \frac{5}{4}h \implies h = 40

Cross-multiplying and isolating

h

yields the correct height of the cliff in feet.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)

Estimated Time:1m 30s

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