Question

The shadow of a lamp post is `3sqrt3` m long when the angle of elevation of the sun is 60°. Arushi is of height 3 m standing in front on the lamp post.

1. What is the height of the lamp post?
2. What is the length of shadow of Arushi?
3. How far is Aarushi from the lamp post?

Answer 1

Let the height of the lamp post be h, and let the total length of the shadow of the lamp post be given as \[3\sqrt{3}\text{ m}\] with the angle of elevation of the sun being 60°.

(i)

In the right-angled triangle formed by the lamp post and its shadow:

\[\tan(60^\circ) = \frac{\text{Height of the lamp post}}{\text{Total shadow length}}\]

\[\sqrt{3} = \frac{h}{3\sqrt{3}}\]

\[h = \sqrt{3} \times 3\sqrt{3} = 3 \times 3 = 9\text{ m}\]

Thus, the height of the lamp post is 9 m.

(ii)

Let the length of Arushi's shadow be x. Arushi's height is given as 3 m, and the angle of elevation of the sun is 60°.

\[\tan(60^\circ) = \frac{\text{Height of Arushi}}{\text{Length of Arushi's shadow}}\]

\[\sqrt{3} = \frac{3}{x}\]

\[x = \frac{3}{\sqrt{3}} = \sqrt{3}\text{ m}\]

Thus, the length of Arushi's shadow is \[\sqrt{3}\text{ m}\] (or approximately 1.732 m).

(iii)

From the diagram, the distance between Arushi and the lamp post is the total shadow length minus the length of Arushi's shadow:

\[\text{Distance} = 3\sqrt{3} - \sqrt{3} = 2\sqrt{3}\text{ m}\]

Thus, Arushi is \[2\sqrt{3}\text{ m}\] (or approximately 3.464 m) away from the lamp post.