Question

From a point P on the ground, the angle of elevation of a 10m tall building is \[{{30}^{o}}\]. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is \[{{45}^{o}}\]. Find the length of the flagstaff and the distance of the building from point P.

Answer 1

Hint: First of all, draw a building AB of 10m and its angle of elevation at point P as \[{{30}^{o}}\]. Now draw a flagstaff CA and angle of elevation of its top at point P as \[{{45}^{o}}\]. Now take \[\tan {{30}^{o}}\text{ and }\tan {{45}^{o}}\] and use \[\tan \theta =\dfrac{perpendicular}{base}\] on these two angles to get the required values.

Complete step-by-step answer:
In this question, we are given that from a point P on the ground the angle of elevation of a 10m tall building is \[{{30}^{o}}\]. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is \[{{45}^{o}}\]. Here, we have to find the length of the flagstaff and the distance of the building from point P.
First of all, let us draw a 10m long building AB whose angle of elevation from point P on the ground is \[{{30}^{o}}\].
Now, let us draw a flagstaff CA on top of the building whose angle of elevation at point P is \[{{45}^{o}}\].

Let us consider the length of the flagstaff CA = h and distance of the building from point P, i.e. BP = d. We know that,
\[\tan \theta =\dfrac{perpendicular}{base}\]
In triangle ABP,
\[\tan \left( \angle APB \right)=\dfrac{AB}{BP}....\left( i \right)\]
We are given that,
\[\angle APB={{30}^{o}}\]
\[AB=10m\]
\[BP=d\]
By substituting these values in equation (i), we get,
\[\tan {{30}^{o}}=\dfrac{10}{d}....\left( ii \right)\]
In triangle CBP,
\[\tan \left( \angle CPB \right)=\dfrac{CB}{BP}....\left( iii \right)\]
We are given that,
\[\angle CPB={{45}^{o}}\]
\[CB=CA+AB=h+10m\]
\[BP=d\]
By substituting these values in equation (iii), we get,
\[\tan {{45}^{o}}=\dfrac{h+10}{d}....\left( iv \right)\]
By dividing equation (ii) by equation (iv), we get,
\[\dfrac{\tan {{30}^{o}}}{\tan {{45}^{o}}}=\dfrac{\dfrac{10}{d}}{\dfrac{h+10}{d}}\]
\[\dfrac{\tan {{30}^{o}}}{\tan {{45}^{o}}}=\left( \dfrac{10}{d} \right)\left( \dfrac{d}{h+10} \right)\]
By canceling the like terms, we get,
\[\dfrac{\tan {{30}^{o}}}{\tan {{45}^{o}}}=\dfrac{10}{h+10}....\left( v \right)\]
Now, let us find the value of \[\tan {{30}^{o}}\text{ and }\tan {{45}^{o}}\] from the trigonometric table of general angles.

	\[\sin \theta \]	\[\cos \theta \]	\[\tan \theta \]	\[\operatorname{cosec}\theta \]	\[\sec \theta \]	\[\cot \theta \]
0	0	1	0	-	1	-
\[\dfrac{\pi }{6}\]	\[\dfrac{1}{2}\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{\sqrt{3}}\]	2	\[\dfrac{2}{\sqrt{3}}\]	\[\sqrt{3}\]
\[\dfrac{\pi }{4}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{1}{\sqrt{2}}\]	1	\[\sqrt{2}\]	\[\sqrt{2}\]	1
\[\dfrac{\pi }{3}\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{2}\]	\[\sqrt{3}\]	\[\dfrac{2}{\sqrt{3}}\]	2	\[\dfrac{1}{\sqrt{3}}\]
\[\dfrac{\pi }{2}\]	1	0	-	1	-	0

From the above table, we get,
\[\tan {{30}^{o}}=\dfrac{1}{\sqrt{3}}\]
\[\tan {{45}^{o}}=1\]
By substituting these values in equation (v), we get,
\[\dfrac{\dfrac{1}{\sqrt{3}}}{1}=\dfrac{10}{h+10}\]
\[\dfrac{1}{\sqrt{3}}=\dfrac{10}{h+10}\]
By cross multiplying the above equation, we get,
\[h+10=10\sqrt{3}\]
\[h=10\sqrt{3}-10\]
\[h=10\left( \sqrt{3}-1 \right)m\]
We know that,
\[\sqrt{3}\simeq 1.732\]
So, we get,
\[h=10\left( 1.732-1 \right)\]
\[h=10\left( 0.732 \right)\]
\[h=7.32m\]
Now considering equation (ii),
\[\tan {{30}^{o}}=\dfrac{10}{d}\]
We know that, \[\tan {{30}^{o}}=\dfrac{1}{\sqrt{3}}\]. So, we get,
\[\dfrac{1}{\sqrt{3}}=\dfrac{10}{d}\]
\[d=10\sqrt{3}m\]
\[d=10\left( 1.732 \right)\]
\[d\approx 17.32m\]
Hence, we get the length of the flagstaff as 7.32m and distance between the building and point P is 17.32m.
Note: In this question, students often make the mistake of taking \[\angle APC={{45}^{o}}\] which is wrong because it is written that from the top of the flagstaff, the angle of elevation is \[{{45}^{o}}\] at point P and not that flagstaff is subtending angle \[{{45}^{o}}\] at point P. So, this must be taken care of and students should properly read the question and draw the diagram first to visualize these types of questions.