Question

A ladder $9{\text{m}}$ long reaches a point $9{\text{m}}$ below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the flagstaff is ${60^ \circ }$. What is the height of the flagstaff?
A.$9{\text{m}}$
B.$10.5{\text{m}}$
C.$13.5{\text{m}}$
D.$15{\text{m}}$

Answer 1

Hint: Here, the total length of the flagstaff (BC) will be the sum of the length from the top of the flagstaff to the point $9{\text{m}}$ below the top of a vertical flagstaff (DC) and the length below this point (BD). So here since the angle of elevation is bisected by the ladder in half then $\angle BAD = {30^ \circ }$. Then use the sine ratio $\sin \theta = \dfrac{P}{H}$ where P is perpendicular and H is hypotenuse in triangle ABD to find the value of BD and then add it to CD to get the answer.

Complete step-by-step answer:
Let AD be the ladder whose length=$9{\text{m}}$ and it reaches at point D which is $9{\text{m}}$below the top of a vertical flagstaff BC. It is given that $\angle BAC = {60^ \circ }$ which means $\angle BAD = \angle CAD = {30^ \circ }$
We have to find BC. Let BD=h cm so we can say that BC=BC+DC-- (i)
Now we have to find BD.

In triangle ABD, $\sin \theta = \dfrac{{BD}}{{AD}}$ because we know that $\sin \theta = \dfrac{P}{H}$ where P is perpendicular and H is hypotenuse.
On putting the given values, we get-
$ \Rightarrow \sin {30^ \circ } = \dfrac{h}{9}$
We know that $\sin {30^ \circ } = \dfrac{1}{2}$
So on putting this value, we get-
$ \Rightarrow \dfrac{1}{2} = \dfrac{h}{9}$
On rearranging, we get-
$ \Rightarrow h = \dfrac{9}{2} = 4.5$m
On putting this value and value of CD in eq. (i), we get-
$ \Rightarrow $ BC=$4.5 + 9 = 13.5$ m
The correct answer is option C.
Note: Here, in triangle ABD we have not taken the angle as $\cos \theta $ or $\tan \theta $ because in both their ratios the base comes in the denominator, and in triangle ABD, we do not know the value of the base AB.

Here $\cos \theta = \dfrac{B}{H} = \dfrac{{AB}}{{AD}}$ but here we do not know the value of AB and we have to find BC which cannot be found using this.
When we are solving such a question, we always use the angle which can be used to find the unknown value using the known value like- here in the question, we have used the known value (AD) to get the value of BD.