Question

A vertical stick 12 m long casts a shadow 8 m long on the ground. At the same time, a tower casts a shadow of length 40 m on the ground . Determine the height of the tower.

Answer 1

Let AB represent the height of the vertical stick and PQ represent the height of the tower.

Given: AB=12 m , BC=8 m and QR=40 m

$$\begin{gathered} \text{In}△\mathrm{ABC}\text{and}△\mathrm{PQR}, \\ \angle \mathrm{ABC}=\angle \mathrm{PQR}[90°\mathrm{each}] \\ \angle \mathrm{ACB}=\angle \mathrm{PRQ}[\mathrm{Angle}\mathrm{of}\mathrm{elevation}\mathrm{would}\mathrm{be}\mathrm{same}\mathrm{for}\mathrm{both}\mathrm{the}\mathrm{cases}] \\ \mathrm{Therefore},\mathrm{by}\mathrm{AA}\mathrm{similarity}\mathrm{test}, \\ △\mathrm{ABC}~△\mathrm{PQR} \\ \mathrm{Hence},\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}}=\frac{\mathrm{CA}}{\mathrm{RP}}[\mathrm{Corresponding}\mathrm{sides}\mathrm{of}\mathrm{similar}\mathrm{triangles}] \\ \mathrm{Considering}\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}} \\ \Rightarrow \frac{12}{\mathrm{PQ}}=\frac{8}{40}\Rightarrow \mathrm{PQ}=\frac{12\times 40}{8}=60\mathrm{m} \end{gathered}$$

$\mathrm{Therefore},\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{tower}\mathrm{is}60\mathrm{m}.$