Question

A boy standing X meters behind and opposite the centre of a football goal post observes the angle of elevation of the nearer cross bar as alpha and father cross bar as beta. Show that the length of field is x( tan alpha cot beta -1)meters

Answer 1

see the attachment for your answer.

Answer 2

$L = x(\tan \alpha \cot \beta - 1)$ Proved.

Step-by-step explanation:

See the attached diagram.

Let, AB and CD are the crossbars of the two-goal post and O is the position of the boy.

Hence, OA = x meters and AC = L meters (say)

Now, ∠ BOA = $\alpha$ and ∠ DOC = $\beta$ {Given}

From, Δ OAB, we get

So, $\tan \alpha = \frac{AB}{OA} = \frac{H}{x}$

⇒ $H = x \tan \alpha$ ........ (1)

Again, from Δ OCD, we get

$\tan \beta = \frac{H}{x + L} = \frac{x \tan \alpha}{x + L}$ {From equation (1)}

⇒ $x + L = x \tan \alpha \cot \beta$

⇒ $L = x(\tan \alpha \cot \beta - 1)$ {Proved}