Question

The lateral surface area of a hollow cylinder is 4224 cm². It is cut along its height and formed a rectangular sheet of width 32 cm. Find the perimeter and area of the area of the rectangular sheet.​

Answer 1

Step-by-step explanation:

$\large \red{ \frak{ \sf \frak{✇ \: Given: - }}}$

• Lateral Surface Area is 4224 cm².

• Width of sheet = 32 cm.

$\large \red{ \frak{ \ \frak{✇ \:To \: Find: - }}}$

• The perimeter of the sheet

• The area of the sheet

$\large \red{ \frak{ \ \frak{✇ \:Solution: - }}}$

$\sf \: \bigstar \: We \: have -$

• h = Height of the cylinder = Width of the sheet = 32 cm

• Lateral surface are of the cylinder = 4224 cm²

$\sf \bigstar \: Let \: -$

• r be the radius of the base of the hollow cylinder.

So,

ACQ,

$\tt \: \therefore \: 2\pi rh = 4224 \\ \sf \: ➯ \: 2 \times \frac{22}{7} \times r \times 32 = 4224 \\ ➯ \sf \: r = \cancel \frac{4224 \times 7}{2 \times 22 \times 32} \\ \\ \sf \: ➯ \: \large \blue{\frak{r = 21 \: cm}}$

Now,

$\sf \: Length _{(sheet)} = 2\pi r \\ ➥ \sf \: Length _{(sheet)} = 2 \times \frac{22}{ \cancel7} \times \cancel{21} \\ ➥ \sf \: Length _{(sheet)} = 2 \times 22 \times 3 \\ ➥ \bf \: \boxed{ \bf Length _{(sheet)} =132 \: cm }$

So,

$\sf \: Perimeter_{(sheet)} = 2(l + b) \\ ⇰ \sf \: Perimeter_{(sheet)} = 2(132 + 32) \\ ⇰\sf \: Perimeter_{(sheet)} = 2 \times 164 \\ ⇰\bf \pink \large \frak{ \large \: \purple{ \boxed{ \green{ \frak{Perimeter_{(sheet) = \large \: 328 \: cm}}}}}}$

Also,

$\sf \: Area _{(sheet)} = l \times b \\ ➛ \sf \: Area _{(sheet)} = 132 \times 32 \\ ➛ \sf \frak{ \pink{\boxed{ \large{ \green{\: \frak{Area _{(sheet)} = 4224 \: {cm}^{2} }}}}}}$

$\tiny \color{fuchsia}\underline{\rm \therefore The\: Area_{(sheet)} \:and \:Perimeter_{(sheet)} \:is\: 4224 \:cm² \:and \:328 \:cm\: respectively. }$

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