A solid wooden cylinder is of radius 6 cm and height 16 cm. Two cones each of radius 2 cm and height 6 cm are drilled out of the cylinder. Find the volume of the remaining solid. Take π = 22 7 ​

[tex]\large\underline{\sf{Solution-}}[/tex]

Dimensions of solid wooden cylinder

Radius, r = 6 cm

Height, h = 16 cm

Dimensions of cone

Radius, R = 2 cm

Height, H = 6 cm

According to statement, a solid wooden cylinder is of radius 6 cm and height 16 cm. Two cones each of radius 2 cm and height 6 cm are drilled out of the cylinder.

[tex]\bf \: \: Volume_{(Remaining\:solid)} \\ \\ [/tex]

[tex]\sf \: = \: Volume_{(Cylinder)} \: - \: 2 \: Volume_{(Cone)} \\ \\ [/tex]

[tex]\sf \: = \: \pi {(r)}^{2} h \: - \: 2 \times \dfrac{1}{3} \: \pi {(R)}^{2} H \\ \\ [/tex]

[tex]\sf \: = \: \pi {(r)}^{2} h \: - \: \dfrac{2}{3} \: \pi {(R)}^{2} H \\ \\ [/tex]

[tex]\sf \: = \: \pi \bigg({(r)}^{2} h \: - \: \dfrac{2}{3} \: {(R)}^{2} H\bigg) \\ \\ [/tex]

[tex]\sf \: = \: \pi \bigg({(6)}^{2} (16) \: - \: \dfrac{2}{3} \: {(2)}^{2} (6)\bigg) \\ \\ [/tex]

[tex]\sf \: = \: \pi \bigg(36 \times 16 \: - \: 16\bigg) \\ \\ [/tex]

[tex]\sf \: = \: 16 \: \pi \bigg(36 - 1\bigg) \\ \\ [/tex]

[tex]\sf \: = \: 16 \times \frac{22}{7} \times 35 \\ \\ [/tex]

[tex]\sf \: = \: 16 \times 22 \times 5\\ \\ [/tex]

[tex]\sf \: = \: 16 \times 110\\ \\ [/tex]

[tex]\sf \: = \: 1760 \: {cm}^{3} \\ \\ [/tex]

Hence,

[tex]\sf \:\bf\implies \:Volume_{(Remaining\:solid)} = \: 1760 \: {cm}^{3} \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: Formulae}}}} \\ \\ \bigstar \: \bf{CSA_{(cylinder)} = 2\pi \: rh}\\ \\ \bigstar \: \bf{Volume_{(cylinder)} = \pi {r}^{2} h}\\ \\ \bigstar \: \bf{TSA_{(cylinder)} = 2\pi \: r(r + h)}\\ \\ \bigstar \: \bf{CSA_{(cone)} = \pi \: r \: l}\\ \\ \bigstar \: \bf{TSA_{(cone)} = \pi \: r \: (l + r)}\\ \\ \bigstar \: \bf{Volume_{(sphere)} = \dfrac{4}{3}\pi {r}^{3} }\\ \\ \bigstar \: \bf{Volume_{(cube)} = {(side)}^{3} }\\ \\ \bigstar \: \bf{CSA_{(cube)} = 4 {(side)}^{2} }\\ \\ \bigstar \: \bf{TSA_{(cube)} = 6 {(side)}^{2} }\\ \\ \bigstar \: \bf{Volume_{(cuboid)} = lbh}\\ \\ \bigstar \: \bf{CSA_{(cuboid)} = 2(l + b)h}\\ \\ \bigstar \: \bf{TSA_{(cuboid)} = 2(lb +bh+hl )}\\ \: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]

Answer:

The correct answer of this question is [tex]1760cm^{3}[/tex]

Step-by-step explanation:

Given - A solid wooden cylinder is of radius 6 cm and height 16 cm. Two cones each of radius 2 cm and height 6 cm are drilled out of the cylinder.

To Find - Find the volume of the remaining solid.

Radius a solid wooden cylinder is  r = 6 cm

Height a solid wooden cylinder is h = 16 cm

A solid wooden cylinder with a radius of 6 cm and a height of 16 cm has been created. The cylinder is drilled with two cones, each with a radius of 2 cm and a height of 6 cm.

[tex]Volume_{cylinder} - 2 Volume_{cone} \\\pi r^{2}h - 2 . \frac{1}{3} \pi R^{2} H\\\pi r^{2} h - \frac{2}{3} \pi R^{2} H\\\pi (r^{2} h - \frac{2}{3} 2^{2} . 6)\\\pi (36 . 16 - 16 )\\16\pi ( 36 - 1)\\16 . \frac{22}{7} . 35\\16 . 22 . 5\\16 . 110 = 1760cm^{3}[/tex]

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