Question

Diameter of cylinder A is 7 cm,
and the height is 14 cm. Diameter of cylinder
Bis 14 cm and height is 7 cm. Without doing
any calculations can you suggest whose volume
is greater ? Verify it by finding the volume of
both the cylinders. Check whether the cylinder
with greater volume also has greater surface
area ?​

Answer 1

$\huge{\underline{\underline{\sf{\purple{Required\:Solution:}}}}}$

Volume of cylinder B is greater.

For Cylinder A

$\displaystyle{\sf{\:\:\:\:\:\:\:\:r(radius)\:=\: \dfrac{7}{2}cm}}$

$\displaystyle{\sf{\:\:\:\:\:\:\:\:h(height)\:=\: 14cm}}$

$\displaystyle{\boxed{\sf{ Volume\:=\: \pi r^{2}h}}}$

• Substitute the values and simplify it

$\displaystyle{\sf{\:\:\:\:\:\:\:\:\:\:= \dfrac{22}{7} \times \bigg(\frac{7}{2}^{2} \bigg)\times14}}$

$\large{\boxed{\sf{\orange{= 539 cm^{3}}}}}$

For Cylinder B

$\displaystyle{\sf{\:\:\:\:\:\:\:\:r(radius)\:=\: \dfrac{14}{2}cm=7cm}}$

$\displaystyle{\sf{\:\:\:\:\:\:\:\:h(height)\:=\: 7cm}}$

$\displaystyle{\boxed{\sf{ Volume\:=\: \pi r^{2}h}}}$

• Substitute the values and simplify it.

$\displaystyle{\sf{\:\:\:\:\:\:\:\:\:\:= \dfrac{22}{7} \times (7^{2})\times7}}$

$\large{\boxed{\sf{\orange{= 1078 cm^{3}}}}}$

By actual calculation of volumes of both the cylinders, it is verified that the volumes of cylinder B is greater.

For Cylinder A

$\displaystyle{\boxed{\sf{ \:\:\:Surface\:area\:=\:2 \pi r(r+h)}}}$

• Substitute the values and simplify it

$\displaystyle{\sf{\:\:\:\:\:\:\:\:\:\:\:\:\:\:= 2 \times \dfrac{22}{7} \times \dfrac{7}{2} \times \bigg( \dfrac{7}{2} + 14\bigg)}}$

$\displaystyle{\sf{\:\:\:\:\:\:\:\:\:\:\:\:\:\:= 2 \times \dfrac{22}{7} \times \dfrac{7}{2} \times \dfrac{35}{2}}}$

$\large{\boxed{\sf{\pink{=\:385\:cm^{2}}}}}$.

For Cylinder B

$\displaystyle{\boxed{\sf{ \:\:\:Surface\:area\:=\:2 \pi r(r+h)}}}$

• Substitute the values and simplify it.

$\displaystyle{\sf{\:\:\:\:\:\:\:\:\:\:\:\:\:\:= 2 \times \dfrac{22}{7} \times 7\times (7+7)}}$

$\displaystyle{\sf{\:\:\:\:\:\:\:\:\:\:\:\:\:\:= 2 \times \dfrac{22}{7} \times 7 \times 14}}$

$\large{\boxed{\sf{\pink{=\:616\:cm^{2}}}}}$.

By actual calculation of surface area of both the cylinders, we observed that the cylinder with greater volume has greater surface area.

And we are done! :D

__________________________

Answer 2

Volume of Cylinder A = πr²h

⇒ Volume of Cylinder A = π × r² ×h

⇒ Volume of Cylinder A = π × (diameter/2)² × h

⇒ Volume of Cylinder A = 22/7 × (7/2)² × h

⇒ Volume of Cylinder A = 22/7 ×(7/2)² × 14

⇒ Volume of Cylinder A = 22/7 × (7/2)² × 14

⇒ Volume of Cylinder A = 22/7 × 49/4 ×14

⇒ Volume of Cylinder A = 22/1 × 49/4 × 2

⇒ Volume of Cylinder A = 22 × 49/4 × 2

⇒ Volume of Cylinder A = 22 × 49/2 × 1

⇒ Volume of Cylinder A = 22 × 49/2

⇒ Volume of Cylinder A = 11 × 49/1

⇒ Volume of Cylinder A = 11 × 49

⇒ Volume of Cylinder A = 539 cm³

       

Volume of Cylinder B = πr²h

⇒ Volume of Cylinder B = π × r² ×h

⇒ Volume of Cylinder B = π × (diameter/2)² × h

⇒ Volume of Cylinder B = 22/7 × (14/2)² × h

⇒ Volume of Cylinder B = 22/7 × (7)² × h

⇒ Volume of Cylinder B = 22/7 × 49 × h

⇒ Volume of Cylinder B = 22/1 × 7 × h

⇒ Volume of Cylinder B = 22 × 7 × h

⇒ Volume of Cylinder B = 22 × 7 × 7

⇒  Volume of Cylinder B  = 1078 cm³

Surface Area of Cylinder A  = 2πr(h + r)

⇒ Surface Area of Cylinder A  = 2 × π × (diameter/2)[(h + (diameter/2))]

⇒ Surface Area of Cylinder A  = 2 × 22/7 × (diameter/2)[(h + (diameter/2))]

⇒ Surface Area of Cylinder A  = 2 × 22/7 × 7/2[(h + (7/2))]

⇒ Surface Area of Cylinder A  = 1 × 22/7 × 7/1[(h + (7/2))]

⇒ Surface Area of Cylinder A  = 22/1 × 1 [(h + (7/2))]

⇒ Surface Area of Cylinder A  = 22 [(h + (7/2))]

⇒ Surface Area of Cylinder A  = 22 [(14 + (7/2))]

⇒ Surface Area of Cylinder A  = 22 [(28/2 + 7/2)]

⇒ Surface Area of Cylinder A  = 22 (35/2)

⇒ Surface Area of Cylinder A  = 22 × (35/2)

⇒ Surface Area of Cylinder A  = 11 × (35/1)

⇒ Surface Area of Cylinder A  = 11 × 35

⇒ Surface Area of Cylinder A  = 385 cm²

Surface Area of Cylinder B  = 2πr(h + r)

⇒ Surface Area of Cylinder B  = 2 × π × (diameter/2)[(h + (diameter/2))]

⇒ Surface Area of Cylinder B  = 2 × 22/7 × (diameter/2)[(h + (diameter/2))]

⇒ Surface Area of Cylinder B  = 2 × 22/7 × (14/2)[(7 + (14/2))]

⇒ Surface Area of Cylinder B  = 2 × 22/7 × 7 [7 + 7]

⇒ Surface Area of Cylinder B  = 2 × 22/7 × 7 [14]

⇒ Surface Area of Cylinder B  = 2 × 22/1 × 1 [14]

⇒ Surface Area of Cylinder B  = 2 × 22 × 14

⇒ Surface Area of Cylinder B  = 616 cm²

Hence Volume and Surface Area of Cylinder B is Bigger