Math, asked by woodstock68, 2 months ago

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! )​

Answers

Answered by ShírIey
165

Let the side of the sqaure be a cm.

⠀⠀⠀

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

⠀⠀⠀

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

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

Therefore,

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

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

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

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

⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀

I) Finding radius of the circle formed.

  • Circumference of the circle = Length of the wire.

  • Length of the wire is 88 cm.

Now,

\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

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

⠀⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀⠀⠀

⠀⠀⠀⠀⠀

(II) Finding area of the circle.

⠀⠀⠀

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

⠀⠀⠀

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

Therefore,

\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

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

Answered by Anonymous
123

✰ 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 }}

━━━━━━━━━━━━━━━━━━━━━━

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}}

━━━━━━━━━━━━━━━━━━━━━━

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 }}

━━━━━━━━━━━━━━━━━━━━━━

ʀᴇʟᴀᴛᴇᴅ ғᴏʀᴍᴜʟᴀ:

SQUARE :

❏Perimeter= 4a units

❏Area = a² sq.units

❏Volume = a³ cu.units

CIRCLE :

❏Circumference = 2πr units

❏Area = π r² sq.units

Similar questions