Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height \(h\). At a point on the plane, the angle of elevation of the bottom of the flagstaff is \(\alpha\) and that of the top of the flagstaff is \(\beta\). Prove that the height of the tower is \(\dfrac{h\tan\alpha}{\tan\beta-\tan\alpha}\).

Hint:

In height and distance problems, first assign variables carefully, then form separate \(\tan\theta\) equations for each angle of elevation and eliminate the common horizontal distance.

Answer 1

Step 1: Assume the height of the tower and horizontal distance.}
Let the height of the tower be \(H\) and the horizontal distance of the observation point from the foot of the tower be \(x\).
The height of the flagstaff is already given as \(h\).
Therefore, the total height up to the top of the flagstaff is:
\[ H+h \]
Step 2: Form the trigonometric equation for the bottom of the flagstaff.}
The bottom of the flagstaff is the top of the tower.
Since the angle of elevation of this point is \(\alpha\), we have:
\[ \tan\alpha=\frac{\text{height of tower}}{\text{horizontal distance}} \] \[ \tan\alpha=\frac{H}{x} \] So,
\[ H=x\tan\alpha \]
Step 3: Form the trigonometric equation for the top of the flagstaff.}
The angle of elevation of the top of the flagstaff is \(\beta\).
Hence,
\[ \tan\beta=\frac{H+h}{x} \]
Step 4: Substitute the value of \(H\) from Step 2.}
Substituting \(H=x\tan\alpha\) into the second equation, we get:
\[ \tan\beta=\frac{x\tan\alpha+h}{x} \] \[ \tan\beta=\tan\alpha+\frac{h}{x} \]
Step 5: Find the value of \(x\).}
Rearranging the above equation:
\[ \tan\beta-\tan\alpha=\frac{h}{x} \] \[ x=\frac{h}{\tan\beta-\tan\alpha} \]
Step 6: Find the height of the tower.}
We know that:
\[ H=x\tan\alpha \] Substituting the value of \(x\):
\[ H=\frac{h}{\tan\beta-\tan\alpha}\cdot \tan\alpha \] \[ H=\frac{h\tan\alpha}{\tan\beta-\tan\alpha} \]
Step 7: State the required result.}
Hence, the height of the tower is:
\[ \boxed{\frac{h\tan\alpha}{\tan\beta-\tan\alpha}} \]