Question

A car has two wipers which do not overlap. Each wiper has a blade of length 21cm sweeping through an angle\[{\rm{12}}0^\circ \]. Find the total area cleaned at each sweep of the blades.

Answer 1

Hint: To solve this question first we will have to determine which shape will form when each blade will be swept through an angle of \[{\rm{12}}0^\circ \]and then we will determine the area of the shape formed.

Complete step-by-step answer:
In this question, it is given that the blades are swept through an angle of \[{\rm{12}}0^\circ \]. Here we are going to assume that the blades have a shape of a straight line and the point through which the blade is rotated is fixed. Here we see that when each blade is rotated by an angle of \[{\rm{12}}0^\circ \]about a fixed point, then a sector of the circle will be formed. A sector of a circle is a portion of a circle enclosed by two radii and an arc. The angle between the two radii is called the central angle. In our case, the central angle for all the blades is \[{\rm{12}}0^\circ \]. So in this question we will calculate the area of the sector and we will multiply it by 2 because there are two sectors formed from two wipers. The formula for the calculation of area of sector of circle is as follows:
\[{\rm{Area of sector = }}\dfrac{1}{2}\, \times \,{r^2}\, \times \,\theta \]
Where r is the radius of the sector. In our case it is equal to the length of the wiper .B is the angle between the two radii. In our case, this angle is the angle of sweeping which is equal to \[{\rm{12}}0^\circ \].Therefore, we get
Area swept by one blade \[{\rm{ = }}\,\dfrac{1}{2}\,{\left( {21} \right)^2} \times \,\dfrac{{2\pi }}{3}\]

\[{\rm{ = }}\,\dfrac{1}{2}\,21\, \times \,21\, \times \,\dfrac{2}{3}\, \times \,\dfrac{{22}}{7}\]

\[{\rm{ = }}\,462\,c{m^2}\]

Therefore, the total area will be \[{\rm{ = }}\,{\rm{2}}\, \times {\rm{462}}\,c{m^2}\]

                                     \[{\rm{ = }}\,924\,c{m^2}\]

Note: We have assumed in the question that the wiper is a straight line with no width. If there is a width given, then we have to consider the area of the wiper also.