Question

4. A student has rectangular sheet of dimension 14cm x 22cm. He wants to make a cyanda
such a way that its volume is minimum. Find the height of the cylinder and the minimum
volume​

Answer 1

Question :-

Given above ↑

Answer :-

$\to \boxed{ \sf \: v = 539 \: {cm}^{3} }$

Hence ,volume of cylinder is 539 cm³ ,and its height is 22 cm .

To Find :-

→ Height of the cylinder and minimum volume .

Explanation :-

Given that

• Length of the paper (l) = 14 cm

• Breadth of the paper (b) = 22 cm

• If he wanna make it a cylinder so fold the paper around its breadth , so the length of paper will become height of cylinder .
• And the breadth of paper will become circumference of the cylinder .

Solution :-

According to the condition :

→ Circumference of cylinder = 22cm

→ 2 π r = 22 cm

$\to \sf r = \frac{11}{ \pi} \\ \because \bf \pi = \frac{22}{7} \\ \\ \to \sf r \: = \frac{11 \times 7}{22} \\ \\ \to \boxed{ \sf r = 3.5 \: cm}$

and now ,

$\star \: \sf volume \: of \: cylinder \: (v) = \pi {r}^{2} h \: \\ \\ \to \sf \: v = \frac{22}{7} \times 3.5 \times 3.5 \times 14 \\ \\ \to \sf \: v = 22 \times 3.5 \times 3.5 \times 2 \\ \\ \to \boxed{ \sf \: v = 539 \: {cm}^{3} }$

Hence ,volume of cylinder is 539 cm³ ,

Now ,

• If we fold the paper around its length then height of the cylinder will be equal to 22 cm and its circumference will 14 cm

Now ,using this data

→ circumference = 14 cm

→ 2 π r = 14

→ r = $\frac{7}{\pi}$

hence ,its volume be

→ v = π r² h

→ v = $\frac{22}{7} \times \frac{7}{\pi} \times \frac{7}{\pi} \times 22$

→ v = 343 cm³

So ,volume is minimum at height is 22 cm .