Question

The angle of elevation of a ladder leaning against a wall is ${{60}^{\circ }}$and the foot of the ladder is 4.6 meters away from the wall. The length of the ladder is:
(a) 2.3 meters
(b) 4.6 meters
(c) 7.8 meters
(d) 9.2 meters

Answer 1

Hint: Draw a diagram of the given situation and in the right angle triangle formed use cosine of the given angle to determine the length of ladder. Use the formula: $\cos \theta =\dfrac{\text{Base}}{\text{Hypotenuse}}$. Here, the length of the ladder is hypotenuse and base is the distance of the foot of the ladder from the wall.

Complete step-by-step answer:

Let us draw a diagram of the situation. Let us assume that the height of the wall is $h$ meters and length of the ladder is $l$ meters. When we observe the diagram carefully we will see that the system will form a right angle triangle.

Now, let us come to the question. In the right angle triangle ABC,

Using, $\cos \theta =\dfrac{\text{Base}}{\text{Hypotenuse}}$, we get,

$\cos {{60}^{\circ }}=\dfrac{BC}{AC}$
Here, BC = 4.6 m, AC =$l$ meters and we know that, $\cos {{60}^{\circ }}=\dfrac{1}{2}$.

Therefore,

$\dfrac{1}{2}=\dfrac{4.6}{l}$

By cross-multiplication we get,

$l=2\times 4.6=9.2\text{ m}$

Hence, option (d) is the correct answer.

Note: One can note that in the right angle triangle, we have used cosine of the given angle and not sine or tangent of the given angle. We can use these also but that would be unnecessary because we have been provided with the base and we have to find hypotenuse of the right angle triangle. Therefore, cosine is the best option to solve the question simply and in less time.