Question

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand from 9 am to 9:30 am.​

Answer 1

Answer :-

$\: \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Area of Sectors has been used. In this, for calculating the area formed by a sector of circle, we need to know the radius of the circle and angle θ between the radii forming the sectors. A view of this question's diagram is given in the attachment.

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★ Formula Used :-

$\: \large{\boxed{\boxed{\rm{Area \: of \: sector \: = \: \dfrac{\pi r^{2}\theta}{360^{\circ}}}}}}$

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★ Question :-

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand from 9 am to 9:30 am.

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★ Solution :-

Given,

We know that clock is circular in shape. And area swept by minute hand is like a sector formed. Then the radii will be end to end as a straight lines. That is the angle between them is 180°

So,

» Radii of the circle = Length of minute hand = r = 14 cm

» Angle between the radii = θ = 180°

Now, according to the formula :-

$\: \: \: \\ \large{\rm{\longmapsto \: \; Area \: of \: sector_{(area \: swept \: by \: minute \: hand)} \: = \: \dfrac{\pi r^{2}\theta}{360^{\circ}}}}$

$\: \\ \large{\rm{\longmapsto \: \: Area \: of \: sector_{(area \: swept \: by \: minute \: hand) } \: = \: \dfrac{\dfrac{22}{7} \: \times \: (14)^{2} \: \times \: 180^{\circ}}{360^{\circ}}}}$

$\: \\ \large{\sf{\longmapsto \: \: Area \: of \: sector_{(area \: swept \: by \: minute \: hand) }\: = \: \dfrac{1}{2} \: \times \: 22 \: \times \: 2 \: \times \: 14}}$

➣ Area formed by sector = 22 × 14

= 308 cm²

Since, Area swept by minute hand = Area formed by sector. Then,

$\qquad \qquad \\ \large{\boxed{\tt{\underline{Area \: formed \: by \: sector \: = \: 308 \: cm^{2}}}}}$

$\; \; \\ \large{\boxed{\rm{\leadsto \; \; Thus, \: area \: swept \: by \: minute \: hand \: from \: 9:00 \: am \: \;to\; \: 9:30 \: am \: is \: \boxed{\underline{\bf{308 \: cm^{2}}}}}}}$

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$\: \: \\ \large{\underline{\underline{\bf{\leadsto \: \: Let's \: know \: more \: formulas}}}}$

$\: \\ \Longrightarrow \: \: \rm{Length \: of \: arc \: = \: \dfrac{2\pi r \theta}{360^{\circ}}}$

$\: \: \\ \rm{\Longrightarrow \: \: Area \: subtented \: by \: chord \: = \: \dfrac{\pi r^{2} \theta}{360^{\circ}} \: - \: \dfrac{1}{2} \: \times \: r^{2} \: \times \: \theta}$

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$\: \large{\tt{\underline{\underline{Note \: point \: :-}}}}$

• Here we can find our answer by using the formula of Area of Semi circle. Since we get our value as ½ × πr². But here appropriate method will be using the Area of Sector.

Answer 2

Given :

The length of the minute hand of a clock is 14 cm.

To find :

Find the area swept by the minute hand from 9 am to 9:30 am.​

Solution :

The length of the minute hand of a clock is 14 cm.

∴ Radius (r) = 14 cm

Now in 1 minute,

Area created = 360/60 = 6°

From 9:00 am to 9:30 am = 30 minutes

∴ Area created by 30 minutes = 30 * 6 = 180° = θ

Now,

Area of sector = θ/360° × πr²

⇒  Area swept = 180°/360° × 22/7 * 14²

⇒  Area swept = 1/2 × 22/7 × 196

⇒  Area swept = 11 * 28

⇒  Area swept = 308 cm²

Therefore,

Area swept by minute hand = 308 cm²