Question

A solid right circular cylinder has a total surface area 462 sq. cm. If its curved surface area is one-third of the total surface area, find the volume of this cylinder.
[Use π = 22/7]

Answer 1

GIVEN:-

• TSA of cylinder = 462cm².

• its CSA is one-third of the TSA.

FIND:-

• the volume of this cylinder.

SOLUTION:-

➠ curved surface area = total surface area /3

➠ curved surface area = 462 /3

➠ .°. curved surface area = 154cm.

➠ Rest of area = 462 - 154 = 308 cm.

➠ .°. 2πr² = 308

➠ 2 × 22 / 7 × r² = 308

➠ .°. r² = 308 ×7/44

➠ .°. r² = 49

➠ .°. r = 7 cm.

➠ .°. curved surface area of cylinder = 154cm. (Given)

➠ .°. 2πrh = 154

➠ 2 × 22/7 × 7 ×h =154

➠ .°. h = 154 / 44

➠ .°. h = 3.5cm.

NOW,

➠ volume of cylinder = πr²h

➠ .°. volume of cylinder = 22/7 × (7)² ×3.5

➠ .°. volume of cylinder = 22 × 7 ×3.5

➠ .°. volume of cylinder = 539 cm³.

Answer 2

$\huge \bf Solution$

$\rm \: \: \: \: Let \: the \: base \: radius \: and \: the \: height \: of \: the \: right \: circular \: \\ \rm \: \: \: \: cylinder \: be \: \bf \: r \: \rm cm \: and \: \bf \: h \rm \: cm \: respectively. \: Then,$

$\rm \: \: \: Total \: surface \: area = 462 {}^{2} \: (given)$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm2\pi r {}^{2} h + 2\pi r {}^{2} = \: 462 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ...(1)$

$\: \: \: \rm \: Also, \: given \: that \:$

$\: \: \: \rm \: Curved \: surface \: area$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm = \: \frac{1}{3} \: total \: surface \: area$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm2\pi rh = \: \frac{1}{3}(462) = 154 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ...(2)$

$\rm \: \: \: Putting \: 2\pi rh = 154 \: from \: (2) \: in \: (1)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm154 + 2\pi rh {}^{2} = 462$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm2\pi r {}^{2} \: = \: 462 - 154 = 308$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \rm2 \times \frac{22}{7} r {}^{2} = 308$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm44r {}^{2} \: = \: 7 \times 308$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \: r {}^{2} \: = \: \frac{7 \times 308}{44} \: = \: 7 \times \: \frac{77}{11}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm = \: 7 \times 7 = 7 {}^{2}$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \: r \: = \: 7$

$\rm \: \: \: Putting \: r = 7 \: in \: (2)$

$\: \: \: \rm2 \: \times \: \frac{22}{7} \times 7 \times h = 154$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm2 \times 22 \times h = 154 \: \implies \: h = \: \frac{154}{2 \times 22}$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm \: h = \: \frac{7}{2}$

$\therefore \: \: \: \: \: \: \: \rm \: Volume \: of \: the \: cylinder \:$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm = \: \pi r {}^{2} h$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm = \: \frac{22}{7}(7) {}^{2} \bigg( \frac{7}{2} \bigg)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm = \: \frac{22}{7} \times 7 \times 7 \times \frac{7}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm = \: 11 \times 49 = 539 \: cm {}^{3}$