The lengths of the shadows of a tree of heigh | vedclass

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(D) Let the angles of elevation be $\alpha$, $\beta$, and $\gamma$. The height of the tree is $p$.For a shadow of length $L$, $	an(	heta) = \frac{p}

Vedclass · Accepted Answer

$\frac{\pi}{2}$

Vedclass · Answer

(D) Let the angles of elevation be $\alpha$, $\beta$, and $\gamma$. The height of the tree is $p$.For a shadow of length $L$, $	an(	heta) = \frac{p}{L}$.For shadow $p$: $	an(\alpha) = \frac{p}{p} = 1 \implies \alpha = \frac{\pi}{4}$.For shadow $2p$: $	an(\beta) = \frac{p}{2p} = \frac{1}{2}$.For shadow $3p$: $	an(\gamma) = \frac{p}{3p} = \frac{1}{3}$.We need to find $\alpha + \beta + \gamma = \frac{\pi}{4} + \arctan(\frac{1}{2}) + \arctan(\frac{1}{3})$.Using the formula $\arctan(x) + \arctan(y) = \arctan(\frac{x+y}{1-xy})$:$\arctan(\frac{1}{2}) + \arctan(\frac{1}{3}) = \arctan(\frac{1/2 + 1/3}{1 - 1/6}) = \arctan(\frac{5/6}{5/6}) = \arctan(1) = \frac{\pi}{4}$.Thus, $\alpha + \beta + \gamma = \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2}$.

The lengths of the shadows of a tree of height $p$, thrown by sun's rays at three different moments are $p$, $2p$ and $3p$. The sum of the angles of elevation of the sun's rays at these three moments is equal to...

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