1 Marks Question

Question 11 Mark

The length of the minute hand of a clock is $14\ cm$. Find the area swept by the minute hand in $5$ minutes.

Answer

Here, $r = 14\ cm$ and $\theta = \frac { 90 ^ { \circ } } { 3 } = 30 ^ { \circ }$
$\therefore$Area swept = $\frac { \theta } { 360 ^ { \circ } } \times \pi r ^ { 2 }$
=$\frac { 30 ^ { \circ } } { 360 ^ { \circ } } \times \frac { 22 } { 7 } \times 14 \times 14$
= $\frac { 154 } { 3 } cm^2$

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Question 21 Mark

Area of a sector of angle $p$ (in degrees) of a circle with radius $R$ is

Answer

Area of the sector of angle $p$ of a circle with radius $R$ $= {\theta \over 360} \times \pi r ^2 = {p \over 360} \times \pi R ^2$ $= {p \over 2(360)} \times 2 \pi R ^2 = {p \over 720} \times 2\pi R ^2$

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Question 31 Mark

To warm ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle $80^\circ$ to a distance of $16.5\ km$. Find the area of the sea over which the ships are warned. (use $ \pi = 3.14 )$

Answer

We have, $r = 16.5\ km$ and $\theta$ $= 80^\circ$.
Let $A$ be the area of the sea over which the ships are warmed. Then,
$A = \frac { \theta } { 360 } \times \pi r ^ { 2 }$
$=\frac { 80 } { 360 } \times 3.14 \times 16.5 \times 16.5 \mathrm\; { km } ^ { 2 }$
$= 189.97\ km^2$

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Question 41 Mark

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

Answer

Radius of each wiper $= 25\ cm$, Angle $= 115^\circ$
$\therefore \theta = 115^\circ $
Total area cleaned at each sweep of the blades
$= 2 \left[ \frac { 115 } { 360 } \times \frac { 22 } { 7 } \times 25 \times 25 \right] \left( \because \text { Area } = \frac { \theta } { 360 } \pi r ^ { 2 } \right)$
$= \frac { 230 \times 22 \times 5 \times 25 } { 72 \times 7 }$
$= \frac { 230 \times 11 \times 125 } { 36 \times 7 }$
$= \frac { 115 \times 11 \times 125 } { 18 \times 7 }$
$= \frac { 158125 } { 126 } \mathrm { cm } ^ { 2 }$
$= 1254.96 \mathrm { cm } ^ { 2 }$

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Question 51 Mark

An umbrella has $8$ ribs which are equally spaced (see figure). Assuming umbrella to be a flat circle of radius $45\ cm$, Find the area between the two consecutive ribs of the umbrella.

Answer

Here, $r = 45\ cm$ and $\theta = \frac { 360 ^ { \circ } } { 8 } = 45 ^ { \circ }$
Area between two consecutive ribs of the umbrella = $\frac { \theta } { 360 ^ { \circ } } \times \pi r ^ { 2 }$
= $\frac { 45 ^ { \circ } } { 360 ^ { \circ } } \times \frac { 22 } { 7 } \times 45 \times 45 \quad = \frac { 22275 } { 28 }cm^2$

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Question 61 Mark

Find the area of a sector of a circle with radius $6\ cm$, if the angle of the sector is $60^\circ$.

Answer

We know that Area of sector =$\frac { \theta } { 360 } \pi r ^ { 2 }$
Here, $ \theta = 60 , r = 6$
$\therefore$ Required area $= \frac { 60 } { 360 } \times \frac { 22 } { 7 } \times ( 6 ) ^ { 2 }$
$= \frac { 1 } { 6 } \times \frac { 22 } { 7 } \times 36$
$= \frac { 22 \times 6 } { 7 }$
$= \frac { 132 } { 7 } = 18 \frac { 6 } { 7 } c m ^ { 2 }$

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