Question

The shadow of a flagstaff, $8$ metres in height is $6$ metres. If the shadow of the pole is $15$ metres, find the height of the pole.

Answer 1

Hint:
We will use the method of direct proportion to solve for the height of the pole since the shadows are cast at same interval of time and hence we can use the relation between their original height and their shadows given by: $\dfrac{8}{6} = \dfrac{{{h_p}}}{{15}}$ and by solving this equation, we can easily get the value of height of the pole.

Complete step by step solution:
We are given a flagstaff of height $8m$ and whose shadow is $6m$. There is a pole whose shadow is $15m$ long. We are required to calculate the height of the pole.
Let the height of the pole be ‘${h_p}$’ metres.
Using the method of direct proportion, we can say that the height of the flagstaff (${h_f}$) is proportional to the shadow of flagstaff (${s_f}$) as it will vary according to the height of the flagstaff. Similarly, the height of the pole (${h_p}$) and the shadow of the pole (${s_p}$) will be proportional to each other. Since they were cast in the same conditions, therefore, we can say that: ${h_f}:{s_f}::{h_p}:{s_p}$.
$ \Rightarrow 8:6::{h_p}:15$
And using the formula: $a:b::c:d \equiv \dfrac{a}{b} = \dfrac{c}{d}$, we can write
$ \Rightarrow 8:6::{h_p}:15 \equiv \dfrac{8}{6} = \dfrac{{{h_p}}}{{15}}$
$ \Rightarrow \dfrac{{8 \times 15}}{6} = {h_p}$
$ \Rightarrow {h_p} = \dfrac{{120}}{6} = 20m$

Therefore, the height of the pole is $20m$.

Note:
In this question, you may get confused in drawing the relation of proportionality between the shadows and height of the given objects. We have equated both the proportions of height and shadow of flagstaff and pole because direct proportion is a proportion of two variable quantities when their ratio is constant and here the shadows are being cast under similar circumstances, that’s why we have compared them and thus calculated the height of the pole.