Question

35. A metal wire, when bent in the form of a square of largest area, encloses an area of
484 cm2. Find the length of the wire.
If the same wire is bent to a largest circle, find :

(i) radius of the circle formed.
(ii) area of the circle.


(pls answer fast and correct the fastest and the correct answer will be marked the best! )​

Answer 1

Let the side of the sqaure be a cm.

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$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀

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$\star\:\boxed{\sf{\pink{Area_{\:(square)} = (side)^2}}}$

• Area of the square is given that is 484 cm².

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Therefore,

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$:\implies\sf (side)^2 = Area\\\\\\:\implies\sf a^2 = 484 \\\\\\:\implies\sf a = \sqrt{484} \\\\\\:\implies\sf\pink{a = 22\; cm}$

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To calculate the perimeter of the sqaure formula is given by :

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$\star\:\boxed{\sf{\pink{Perimeter_{\:(square)} = 4 \times a}}}$

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$:\implies\sf Perimeter = 22 \times 4 \\\\\\:\implies\sf Perimeter = 88 \; cm$

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I) Finding radius of the circle formed.

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• Circumference of the circle = Length of the wire.

• Length of the wire is 88 cm.

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Now,

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$\longrightarrow\sf 2\pi r = 88 \\\\\\\longrightarrow\sf 2 \times \dfrac{22}{7} \times r = 88 \\\\\\\longrightarrow\sf r = \dfrac{88 \times 7}{22 \times 2}\\\\\\\longrightarrow\sf r = \dfrac{616}{44}\\\\\\\longrightarrow\sf r = \cancel\dfrac{616}{14}\\\\\\\longrightarrow{\underline{\boxed{\frak{ r = 14 \; cm}}}}\;\bigstar$

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$\therefore{\underline{\sf{Hence,\; required\; radius\; is\; \bf{ 14\;cm}.}}}$

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(II) Finding area of the circle.

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$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀

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$\star\;\boxed{\sf{\purple{Area_{\:(circle)} = \pi r^2}}}$

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Therefore,

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$\longrightarrow\sf Area_{\:(circle)} = \pi r^2 \\\\\\\longrightarrow\sf Area_{\:(circle)} = \dfrac{22}{7} (14)^2 \\\\\\\longrightarrow\sf Area_{\:(circle)} = \dfrac{22}{7} \times 14 \times 14 \\\\\\\longrightarrow{\underline{\boxed{\frak{\purple{Area_{\:(circle)} = 616\;cm^2}}}}}\;\bigstar$

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$\therefore{\underline{\sf{Hence,\; area \; of \; the \; circle\; is\; \bf{ 616\;cm^2}.}}}$

Answer 2

✰ ᴜɴᴅᴇʀsᴛᴀɴᴅɪɴɢ ᴛʜᴇ ǫᴜᴇsᴛɪᴏɴ :

A metal wire is first bent into a square shape and had a area of 484 cm² .Next, the same wire is rebent into a circle shape. We have to find the radius and area of the circle and then we should find the length of the wire .

✰ɢɪᴠᴇɴ:

• Area of the square = 484 cm²

✰ᴛᴏ ғɪɴᴅ:

• length of the wire
• radius of the circle
• area of the circle

✰sᴏʟᴜᴛɪᴏɴ:

☛Length of the wire :

Let the side of the square be x

given that

➠Area of the square = 484 cm²

➠x² = 484 cm²

➠x = √484 cm²

➠$\boxed{\sf{x = 22 cm }}$

✰ɴᴏᴛᴇ:

As the same wire is bent into square and then into circle ,

length of the wire = Perimeter of the square = circumference of the circle

➠Perimeter of the square

➠4x

➠4(22)

➠88cm

therefore,

$\boxed{\sf{length of the wire = 88cm }}$

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☛Radius of the circle :

circumference of the circle = Perimeter of the square

➠2πr = 88

➠πr = 44

➠r = 44 × (7/22)

➠r = 2 × 7

➠r = 14 cm

$\boxed{\sf{Radius of the circle= 14 cm}}$

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☛Area of the circle:

➠area of the circle

➠πr² cm²

➠(22/7) × 14 × 14

➠44 × 14

➠616 cm²

$\boxed{\sf{area of the circle= 616cm² }}$

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✰ʀᴇʟᴀᴛᴇᴅ ғᴏʀᴍᴜʟᴀ:

SQUARE :

❏Perimeter= 4a units

❏Area = a² sq.units

❏Volume = a³ cu.units

CIRCLE :

❏Circumference = 2πr units

❏Area = π r² sq.units