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A cylindrical boiler, 2 m high, is 3.5 m in diameter. It has a hemispherical lid. Find the volume of its interior, including the part covered by the lid.

Mensuration

ICSE

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Answer

Given,

Diameter of cylindrical boiler = 3.5 m

Radius of cylindrical boiler (R) = 3.52=3520=74\dfrac{3.5}{2} = \dfrac{35}{20} = \dfrac{7}{4} m.

Height (H) = 2 m.

From figure,

A cylindrical boiler, 2 m high, is 3.5 m in diameter. It has a hemispherical lid. Find the volume of its interior, including the part covered by the lid. Cylinder, Cone, Sphere, Concise Mathematics Solutions ICSE Class 10.

Radius of hemispherical lid = Radius of cylindrical boiler = R = 74\dfrac{7}{4} m.

Total volume of the boiler = Volume of cylindrical boiler + Volume of hemispherical lid

=πR2H+23πR3=πR2(H+23×R)=227×(74)2×(2+23×74)=227×4916×(2+76)=227×4916×196=20482672=30.48 m3.= πR^2H + \dfrac{2}{3}πR^3 \\[1em] = πR^2(H + \dfrac{2}{3} \times R) \\[1em] = \dfrac{22}{7} \times \Big(\dfrac{7}{4}\Big)^2 \times \Big(2 + \dfrac{2}{3} \times \dfrac{7}{4}\Big) \\[1em] = \dfrac{22}{7} \times \dfrac{49}{16} \times \Big(2 + \dfrac{7}{6}\Big) \\[1em] = \dfrac{22}{7} \times \dfrac{49}{16} \times \dfrac{19}{6} \\[1em] = \dfrac{20482}{672} \\[1em] = 30.48 \text{ m}^3.

Hence, volume of boiler = 30.48 m3.

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