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Two people standing on the same side of a tower in straight li

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Two people standing on the same side of a tower in straight line with it, measure the angles of elevation of the height of tower as $2 5^{0}$ and $5 0^{0}$ respectively if the height of the tower is 70 m, find the distance between the two people.

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Two people standing on the same side of a tower in straight line with it, measure the angles of elevation of the height of tower as

2 5^{0}

and

5 0^{0}

respectively if the height of the tower is 70 m, find the distance between the two people.

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Text solutionVerified

Let

A B

be the tower of height

70 m

Now,

For first person:-

θ = 50°

A B = 70 m

As we know that,

tan θ = \frac{Perpendicular}{Base}

Therefore,

△ A B D

tan 50° = \frac{A B}{B D}

\Rightarrow B D = \frac{70}{tan 50°} ..... (1)

Similarly, for second person:-

θ = 25°

A B = 70 m

△ A BC

tan 25° = \frac{A B}{BC}

\Rightarrow BC = \frac{70}{tan 25°} ..... (2)

Therefore,

Distance between the two people

= BC - B D = \frac{70}{tan 25°} - \frac{70}{tan 50°} = 70 (\frac{1}{tan 25°} - \frac{1}{tan 50°}) = 70 (\frac{1}{tan 25°} - \frac{1 - tan ^{2} 25°}{2 tan 25°}) = \frac{70}{tan 25°} (1 + tan^{2} 25°) = (\frac{70}{tan 25°}) sec^{2} 25° = \frac{70}{sin 25° cos 25°} = \frac{140}{sin 50°}

Hence the distance between the two peope is

\frac{140}{sin 50°}

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Question Text	Two people standing on the same side of a tower in straight line with it, measure the angles of elevation of the height of tower as $2 5^{0}$ and $5 0^{0}$ respectively if the height of the tower is 70 m, find the distance between the two people.
Answer Type	Text solution:1
Upvotes	150