question_answer

A vertical tower is surmounted by a flagstaff of height h metres. At a point on the ground, the angels of elevation of the bottom and top of the flagstaff are \[\alpha \] and \[\beta \] respectively. What is the height of the tower in meters?

A)  \[\frac{h\,\,\tan \,\,\alpha }{\tan \,\,\beta +\tan \,\,\alpha }\]         

B)  \[\frac{h\,\,\tan \,\,\alpha }{\tan \,\,\beta -\tan \,\,\alpha }\]

C)  \[\frac{h\,\,\tan \,\,\alpha }{\tan \,\,\beta }\]                    

D)  \[\frac{h\,\,\tan \,\,\beta }{\tan \,\,\beta -\tan \,\,\alpha }\]

Correct Answer: B

Solution :

(b): In \[\Delta \,ACD,\,\,tan\beta =\frac{x+h}{y}\] \[y=\frac{x+h}{\tan \beta }\]                  ?..(1) In \[\Delta BCD,\tan \alpha =\frac{x}{y}\] \[\therefore y=\frac{x}{\tan \alpha }\]                  ?..(2) From (1) and (2), \[\frac{x+h}{\tan \beta }=\frac{x}{\tan \alpha }\] \[\Rightarrow \]   \[x\,\tan \alpha +h\tan \alpha =x\tan \beta \] \[\Rightarrow \]   \[h\tan \alpha =x(tan\beta -tan\alpha )\] \[\Rightarrow \]   \[x=\frac{h\,\,\tan \alpha }{\tan \beta -\tan \alpha }\]