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The angle of elevation of a ladder leaning against a wall is $ 60^{\circ} $ and the foot of the ladder is $ 9.5 \mathrm{~m} $ away from the wall. Find the length of the ladder.
Given:
The angle of elevation of a ladder leaning against a wall is \( 60^{\circ} \) and the foot of the ladder is \( 9.5 \mathrm{~m} \) away from the wall.
To do:
We have to find the length of the ladder.
Solution:
Let $AB$ be the wall and $AC$ be the ladder.
The foot of the ladder(point $C$) is \( 9.5 \mathrm{~m} \) away from the wall.
From the figure,
$\mathrm{BC}=9.5 \mathrm{~m}, \angle \mathrm{ACB}=60^{\circ}$
Let the length of the ladder be $\mathrm{AC}=h \mathrm{~m}$
We know that,
$\cos \theta=\frac{\text { Base }}{\text { Hypotenuse }}$
$=\frac{\text { BC }}{AC}$
$\Rightarrow \cos 60^{\circ}=\frac{9.5}{h}$
$\Rightarrow \frac{1}{2}=\frac{9.5}{h}$
$\Rightarrow h=9.5 \times 2 \mathrm{~m}$
$\Rightarrow h=19 \mathrm{~m}$
Therefore, the length of the ladder is $19 \mathrm{~m}$.
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