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# The angle of elevation of the top and bottom of a flag staff fixed at the top of a tower at a point distant $a$ ft. from the foot of the tower are $\alpha$ and $\beta$. The height of the tower is ________ A. $a\left( {\tan \beta - \tan \alpha } \right)$B. $a\left( {\tan \beta + \tan \alpha } \right)$C. $a\left( {\cot \beta - \tan \alpha } \right)$D. None of these

Last updated date: 24th Jun 2024
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Hint: Draw the diagram of the given problem statement for a better understanding of the situation. Use the trigonometric ratios, that are $\sin \theta = \dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}}$ and $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$ in the physical triangle formed to find the height of the tower.

Complete step by step answer:
Let us assume the point at a foot distant from the tower be $P$.
We can draw the figure for the given scenario, where the flagstaff is fixed on the top of the tower, which is a foot distant from $P$.
The point $P$ lies in the plane of the bottom of the tower. The angle of elevation from point P to the top of the flag staff is $\alpha$ and angle of elevation from the point $P$ to the bottom of the flag staff is $\beta$.

Here $BC$ represents the height of the tower and $AB$ represents the height of the flag staff.
From the figure, in the triangle $PBC$
$\tan \beta = \dfrac{{BC}}{{PC}}$
Substituting $a$ for $PC$ we get
$\tan \beta = \dfrac{{BC}}{a}$
On simplifying, we get
$a\tan \beta = BC$
Here $BC$ represents the height of the tower, therefore, the height of the tower is $a\tan \beta$.
Thus option D i.e. none of these is the correct answer.

Note: The diagram should be drawn correctly according to the given conditions correctly. The angle of elevation of a higher point is more than that for a lower point in the same vertical line. In a right angled triangle, the $\tan \theta$ is the equal to $\dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$, where perpendicular is the side opposite to the angle $\theta$, and $\sin \theta$ is the equal to $\dfrac{{{\text{Perpendicular}}}}{{{\text{Hypotenuse}}}}$, where perpendicular is the side opposite to the angle $\theta$.