Question

PQ is a post of given height a, and AB is a tower at some distance. If α and β are the angles of elevation of B, the top of the tower, at P and Q respectively. Find the height of the tower and its distance from the post.

Answer 1

$\text{Height of the tower} = \frac{\alpha\ \text{tan}\alpha}{\ \text{tan}\alpha - \text{tan}\beta}$, $\text{Distance from the tower from past is} = \frac{\alpha}{\ \text{tan}\alpha - \text{tan}\beta}$

Step-by-step explanation:

Given :

PQ is the post at height 'a'

Let H be the height of tower AB and x be its distance from PQ

'α' and 'β'  are the angles of elevation of B at P and Q respectively

From the figure , let PA = x PQ = AC = a BC = h AB

$\text{tan} \alpha = \frac{\text{AB}}{\text{AP}}$

$\text{tan} \alpha = \frac{\text{h}}{\text{AP}}$

$\text{AP} = \frac{\text{h}}{\text{tan}\alpha} -----> (1)$

From the right angled triangle BRQ,

$\text{tan} \beta = \frac{\text{BR}}{\text{QR}}$

[Since, AP = RQ]

$\text{tan} \beta = \frac{\text{h} -\alpha}{\text{AP}}$

$\text{AP} = \frac{\text{h}- \alpha}{\text{tan}\beta} -----> (2)$

From (1) and (2)

$\Rightarrow \frac{\text{h}-\alpha}{\text{tan}\beta} = \frac{\text{h}}{\text{tan}\alpha}$

$\Rightarrow {\text{h} \ \text{tan}\alpha-\alpha} {\text{tan}\alpha = \text{h}\ \text{tan}\beta$

$\Rightarrow {\text{h} \ \text{tan}\alpha - \text{h}\ \text{tan}\beta = \alpha} {\text{tan}\alpha$

$\Rightarrow {\text{h} (\ \text{tan}\alpha - \text{tan}\beta )= \alpha} {\text{tan}\alpha$

$\Rightarrow {\text{h} = \frac{\alpha\ \text{tan}\alpha}{\ \text{tan}\alpha - \text{tan}\beta}$

$\text{Height of the tower} = \frac{\alpha\ \text{tan}\alpha}{\ \text{tan}\alpha - \text{tan}\beta}$

Substitute the value of 'h' in Equation (1)

$\text{AP} = \frac{\frac{\alpha\ \text{tan}\alpha}{\ \text{tan}\alpha - \text{tan}\beta}}{\text{tan}\alpha}$

$\text{AP} = \frac{\alpha}{\ \text{tan}\alpha - \text{tan}\beta}$

$\text{Distance from the tower from past is} = \frac{\alpha}{\ \text{tan}\alpha - \text{tan}\beta}$

To Learn More.....

1. The angle of elevation of the top of a tower from two points P and Q at distances of 'a' and 'b' , respectively, from the base and in the same straight line with it are complementary. prove that the height of the tower is √ab

brainly.in/question/2192266

2. Find the angle of elevation of a point which is at a distance of 30 m from the base of a tower 10√3 m high.

brainly.in/question/3025940