Height = \( 10 \cdot \sin(75^\circ) \).

February 22, 2026

$Height = \( 10 \cdot \sin(75^\circ) \).$

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So, \( x^2 - 2x - 8 = 0 \) → factor: \( (x - 4)(x + 2) = 0 \).
So \( x = 4 \) or \( x = -2 \). But \( x > 2 \) for log to be defined, so \( x = 4 \).
A ladder 10 meters long leans against a wall, forming a 75° angle with the ground. How high up the wall does the ladder reach? (Round to nearest tenth)
\( \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \)
= \( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \left( \frac{\sqrt{3} + 1}{2} \right) = \frac{\sqrt{2}(\sqrt{3} + 1)}{4} \approx \frac{1.414 \cdot 2.732}{4} \approx \frac{3.863}{4} \approx 0.9658 \).
So height ≈ \( 10 \times 0.9658 = 9.658 \), rounds to 9.7 meters.

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