# The length of the shadows of a vertical pole of height h, thrown by the sun’s rays at three different moments are h, 2h and 3h.

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The length of the shadows of a vertical pole of height h, thrown by the sun’s rays at three different moments are h, 2h and 3h. The sum of the angles of elevation of the rays at these three moments is equal to:

(a) $\frac{\pi}{4}$

(b) $\frac{\pi}{6}$

(c) $\frac{\pi}{2}$

(d) $\frac{\pi}{3}$

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(c) $\frac{\pi}{2}$

Let PQ be the vertical pole of height h.

If the angle of elevation of the sun’s ray is α, then the length of the shadow QR = h cot α

In ΔQRP, for the first moment, we take α = α1

Given, h cot α1 = h ⇒ cot α1 = 1 ⇒ tan α1 = 1

For the second moment, α = α2 (say)

Given, h cot α2 = 2h ⇒ tan α2$\frac{1}{2}$

Now, for the third moment, α = α3 (say)

h cot α3 = 3h ⇒ tan α$\frac{1}{3}$

Now, we need to find the value of α1 + α2 + α3. tan (α1 + α2 + α3)

tan (α1 + α2 + α3) = tan $\frac{\pi}{2}$ ⇒ α1 + α2 + α$\frac{\pi}{2}$.