Ex.12.2 Q11 Areas Related to Circles Solution - NCERT Maths Class 10

Go back to  'Ex.12.2'

Question

A car has two wipers which do not overlap. Each wiper has a blade of length \(\text{25 cm}\) sweeping through an angle of \(115^\circ. \)Find the total area cleaned at each sweep of the blades.

Text Solution

What is known?

A car has \(2\) wipers which do not overlap. Each wiper has a blade length \(= 25 \,\rm{cm}\) and sweeps through an angle \(\left( \theta \right) = {150^\circ}\)

What is unknown?

Total area cleaned at the sweep of the blade of the \(2\) wipers.

Reasoning:

Visually it is clear that

Area cleaned at the sweep of blades of each wiper \(=\) Area of the sector with angle  \(115^\circ\)at the centre and radius of the circle \(\text{25 cm}\)

Since there are \(2\) wipers of same blade length and same angle of sweeping. Also there is no area of overlap for the wipers.

\(\therefore \;\)Total area cleaned at each sweep of the blades \(=\) \(\begin{align}2\ \times\end{align}\) Area cleaned at the sweep of each wiper.

Steps:

Area cleaned at the sweep of blades of each wiper = Area of the sector of a circle with radius \(\text {25 cm}\) and of angle \(\begin{align}115^{\circ}\end{align}\)

\[\begin{align}& = \frac{\theta }{{{{360}^\circ }}} \times \pi {r^2}\\& = \frac{{{{115}^\circ }}}{{{{360}^\circ }}} \times \pi \times 25 \times 25\\& = \frac{{23}}{{72}} \times 625\pi \end{align}\]

Since there are \(2\) identical blade length wipers

\(\therefore\;\)Total area cleaned at each sweep of the blades

\[\begin{align}& = 2 \times \frac{{23}}{{72}} \times 625\pi\\ & = 2 \times \frac{{23}}{{72}} \times \frac{{22}}{7} \times 625\\& = \frac{{23 \times 11 \times 625}}{{18 \times 7}}\\ &= \frac{{158125}}{{126}}\,{\text{c}}{{\text{m}}^{\text{2}}}\\ & = 1254.96\,\,\,{\text{c}}{{\text{m}}^{\text{2}}}\end{align}\]