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Question: The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sun rays me...

The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sun rays meet the ground at an angle of 600{60^0}. Find the angle between the sun rays and ground at the time of the longer shadow.

A. P=300{\text{P}} = {30^0}
B. P=600{\text{P}} = {60^0}
C. P=450{\text{P}} = {45^0}
D. P=150{\text{P}} = {15^0}

Explanation

Solution

Hint – In order to solve this problem we need to take two triangles and use the concept of trigonometric angles and get the equations to get the angle P using the trigonometric angles itself.

Complete step-by-step answer:

From the given figure AC = x, CD = 2x. Let AB = h and we can clearly see
AD = AC + CD = x + 2x = 3x…………….(1)
In triangle ABC we can say that,

tan60 = ABAC 3 = hx(From(1)) h = x3.......................(2)  \Rightarrow {\text{tan60 = }}\dfrac{{{\text{AB}}}}{{{\text{AC}}}} \\\ \Rightarrow \sqrt {\text{3}} {\text{ = }}\dfrac{{\text{h}}}{{\text{x}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,({\text{From}}\,(1)) \\\ \Rightarrow {\text{h = x}}\sqrt {\text{3}} .......................(2) \\\

In triangle ABD we can say that,
tanP = ABAD tanP = h3x tanP = x33x=13..................(From (2))  \Rightarrow \tan {\text{P = }}\dfrac{{{\text{AB}}}}{{{\text{AD}}}} \\\ \Rightarrow \tan {\text{P = }}\dfrac{{\text{h}}}{{{\text{3x}}}} \\\ \Rightarrow \tan {\text{P = }}\dfrac{{{\text{x}}\sqrt {\text{3}} }}{{{\text{3x}}}} = \dfrac{1}{{\sqrt 3 }}..................({\text{From (2)}}) \\\
So, angle P = 30 degrees.

Hence the right option is A.

Note – To solve this problem we found the relation between h and x and then use the obtained relation in order to calculate angle P here we have used the values that tan60=3\tan 60 = \sqrt 3 and tan30=13\tan 30 = \dfrac{1}{{\sqrt 3 }}. Doing this will solve your problem.