question_answer

A tower subtends an angle\[\alpha \] at a point A in the plane of its base and the angle of depression of the foot of the tower at a point b ft just above A is \[\beta \]. Then, the height of the tower is:

A)  \[b\text{ }tan\text{ }\alpha \text{ }cot\text{ }\beta \]

B)         \[b\text{ }\cot \,\alpha \text{ tan}\,\beta \]

C)  \[b\text{ cot}\,\alpha \text{ }cot\,\beta \]   

D)         \[b\text{ }\tan \,\alpha \text{ tan}\,\beta \]

E)  \[b\text{ ta}{{\text{n}}^{2}}\,\alpha \text{ }cot\,\beta \]

Correct Answer: A

Solution :

Let CD be the tower. Then, from \[\Delta AMC\]                 \[\frac{b}{AC}=\tan \beta \] \[\Rightarrow \]               \[AC=b\cot \beta \] From \[\Delta \Alpha DC,\frac{CD}{AC}=\tan \alpha \] \[\Rightarrow \]               \[CD=b\cot \beta \tan \alpha \]