Brendan was in a boat observing a lighthouse on a vertical cliff at an angle of elevation of $$ 54^{\circ} $$. The cliff was 94 meters high. Find the horizontal distance from Brendan to the base of the cliff. $$ \begin{array}{ll} ext { a) } 56 \mathrm{~m} & ext { go to station } 6 \ ext { b) } 54 \mathrm{~m} & ext { go to station } 8 \ ext { c) } 15 \mathrm{~m} & ext { go to station } 2 \ ext { d) } 68 \mathrm{~m} & ext { go to station } 5 \ ext { e) } 76 \mathrm{~m} & ext { go to station } 4\end{array} $$

Question

Brendan was in a boat observing a lighthouse on a vertical cliff at an angle of elevation of $$ 54^{\circ} $$. The cliff was 94 meters high. Find the horizontal distance from Brendan to the base of the cliff. $$ \begin{array}{ll}	ext { a) } 56 \mathrm{~m} & 	ext { go to station } 6 \ 	ext { b) } 54 \mathrm{~m} & 	ext { go to station } 8 \ 	ext { c) } 15 \mathrm{~m} & 	ext { go to station } 2 \ 	ext { d) } 68 \mathrm{~m} & 	ext { go to station } 5 \ 	ext { e) } 76 \mathrm{~m} & 	ext { go to station } 4\end{array} $$

UpStudy AI · Accepted Answer

To determine the horizontal distance ($$ x $$) from Brendan to the base of the cliff, we can use trigonometric relationships in a right-angled triangle.

Given:
- **Height of the cliff (opposite side)**: $$ 94 $$ meters
- **Angle of elevation**: $$ 54^\circ $$

We can use the **tangent** function, which relates the opposite side to the adjacent side in a right triangle:

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

Substituting the known values:

$$
\tan(54^\circ) = \frac{94}{x}
$$

Solving for $$ x $$:

$$
x = \frac{94}{\tan(54^\circ)}
$$

Calculating $$ \tan(54^\circ) $$ (using a calculator):

$$
\tan(54^\circ) \approx 1.3764
$$

Now, compute $$ x $$:

$$
x = \frac{94}{1.3764} \approx 68.3 \text{ meters}
$$

**Rounded to the nearest meter**, the horizontal distance is **68 meters**.

**Answer:**  
**d) 68 m**
The horizontal distance from Brendan to the base of the cliff is approximately 68 meters.

Brendan was in a boat observing a
lighthouse on a vertical cliff at an angle
of elevation of . The cliff was 94
meters high. Find the horizontal
distance from Brendan to the base of
the cliff.

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Brendan was in a boat observing a lighthouse on a vertical cliff at an angle of elevation of . The cliff was 94 meters high. Find the horizontal distance from Brendan to the base of the cliff.