Question

A solid right cylinder is 18cm
high of its base is 7 cm. Two
equal right cones are drilled on
the plane faces of the cylinder,
the height of each cone being
one - third the height of the cylinder
and the radius of the base of
each cone being equal to the
base of the cylinder, find the
total surface area of the
remaining solid​

Answer 1

AnswEr :

386.76 cm².

$\bf{\purple{\underline{\underline{\tt{Given\::}}}}}$

A solid right cylinder is 18 cm high of it's base is 7 cm. Two equal right cones are drilled on the plane faces of the cylinder, the height of each cone being 1/3 the height of the cylinder and the radius of the base of each cone being equal to the base of the cylinder.

$\bf{\red{\underline{\underline{\tt{To\:find\::}}}}}$

The total surface area of the remaining solid.

$\bf{\green{\underline{\underline{\bf{Explanation\::}}}}}$

$\bf{\large{\underline{\sf{\dag1st\:Case\::\bf{(Cylinder)}}}}}$

$\bf{We\:have}\begin{cases}\sf{Height\:of\:right\:cylider\:(h)=18\;cm}\\ \sf{Radius\:of\:the\:cylinder\:(r)=7\:cm}\end{cases}}$

Formula use :

$\bf{\boxed{\bf{Total\:surface\:area\:of\:cylinder\:=\:2\pi r(r+h)\:\:\:\:(sq.unit)}}}}}}$

$\dashrightarrow\tt{T.S.A.=2\pi r(r+h)}\\\\\\\dashrightarrow\tt{T.S.A.=\bigg[2*\dfrac{22}{\cancel{7}} *\cancel{7}(7+18)\bigg]cm^{2} }\\\\\\\dashrightarrow\tt{T.S.A.=\big(44*25\big)cm^{2} }\\\\\\\dashrightarrow\tt{\red{T.S.A.=1100\:cm^{2} }}$

$\bf{\large{\underline{\sf{\dag2nd\:Case\::\bf{(Right\:Cone)}}}}}$

$\bf{We\:have}\begin{cases}\sf{Height\:of\:right\:cone\:(h)=\frac{1}{\cancel{3}} *\cancel{18}=6\:cm}\\ \sf{Radius\:of\:the\:cone\:(r)=7\:cm}\end{cases}}$

Formula use :

$\bf{\large{\red{Slant\:height\:of\:Cone\:(_{L})=\sqrt{r^{2}+h^{2} }}}} }$

$\leadsto\tt{L=\sqrt{(7)^{2} +(6)^{2} } }\\\\\\\leadsto\tt{L=\sqrt{49+36}\:cm }\\\\\\\leadsto\tt{L=\sqrt{85} \:cm}\\\\\\\leadsto\tt{\red{L=9.21\:cm}}$

Formula use :

$\bf{\large{\boxed{\bf{Total\:surface\:area\:of\:cone=\pi rl+\pi r^{2} }}}}}}$

$\mapsto\tt{T.S.A._{cone}=\pi rl+\pi r^{2} }\\\\\\\mapsto\tt{T.S.A._{cone}=\big[\pi (7)(9.21)+\pi (7)^{2} \big]cm^{2} }\\\\\\\mapsto\tt{T.S.A._{cone}=\big(\pi 64.47+\pi 49\big)cm^{2} }\\\\\\\mapsto\tt{T.S.A._{cone}=\big(113.47\:\pi \big)cm^{2} }\\\\\\\mapsto\tt{T.S.A._{cone}=(113.47*3.14)cm^{2} }\\\\\\\mapsto\tt{T.S.A._{cone}=(2*356.62)\:cm^{2} \:\:\:\:\:\:[2\:right\:cone]}\\\\\\\mapsto\tt{\red{T.S.A._{cone}=713.24\:cm^{2} }}$

Now,

$\bf{\boxed{\bf{The\:total\:surface\:area\:of \:the\:remaining\:solid\:: }}}}}}$

$\bf{\arge{\green{T.S.A._{Cylinder}\:\:-\:\:{\green{T.S.A._{Cone}}}}}}$

$\leadsto\tt{(1100-713.24)cm^{2} }\\\\\\\leadsto\tt{\red{386.76\:cm^{2} }}}$

$\therefore$ $\bf{\large{\orange{\underline{\sf{The\:remaining\:area\:of\:the\:solid\:=386.76\:cm^{2} }}}}}$

Answer 2

Given :

• The height of solid right cylinder = 18 cm.
• Radius of the cylinder = 7 cm.
• The height of right cone = 1/3 × 18 = 6 cm.
• Radius of the cone = 7 cm.

To Find :

• Total surface area of the remaining solid.

______________________________

Case 1 :

Total surface area of cylinder = 2πr(r + h)

$\rightarrow \: \sf{T.S.A. = 2 \pi r(r + h) } \\ \\ \rightarrow \: \sf{T.S.A. = 2 \times \dfrac{22}{ \cancel{7}} \times \cancel{7}(7 + 18) } \\ \\ \rightarrow \: \sf{T.S.A. =44 \times 25} \\ \\ \rightarrow \: \fbox{\sf{T.S.A. =1100 \: {cm}^{2} }}$

Total surface area of cylinder is 1100 cm² .

______________________________

Case 2 :

Total surface area of cone = πrl + πr²

$\qquad \small{ \bf{Slant \: height \: of \: cone \: (l) = \sqrt{ {r}^{2} + {h}^{2} } }} \\ \\ \qquad :\rightarrow\small{ \sf{l = \sqrt{ {7}^{2} + {6}^{2} } }} \\ \\ \qquad : \rightarrow\small{ \sf{l = \sqrt{49 + 36} }} \\ \\ \qquad : \rightarrow\small{ \sf{l = \sqrt{85} }} \\ \\ \qquad : \rightarrow\small{ \bf{l =9.21 \: cm}}$

$\rightarrow \sf{T.S.A. = \pi rl + \pi {r}^{2} } \\ \\ \rightarrow \sf{T.S.A. = \bigg(\dfrac{22}{ \cancel{7}} \times \cancel{7 }\times 9.21} \bigg) + \bigg(\dfrac{22}{ \cancel{7} } \times {7}^{ \cancel{2}} \bigg) \\ \\ \rightarrow \sf{T.S.A. = \big(22 \times 9.21 \big) + \: \big(22 \times 7)} \\ \\ \rightarrow \sf{T.S.A. =202.62 + 154} \\ \\ \rightarrow \sf{T.S.A. =356.62 \: {cm}^{2} }$

For 2 right cones :

→ 2 × 356.62

→ 713.62 cm²

Total surface area of cone = 713.62 cm² .

______________________________

Total surface area of remaining solid

= (T.S.A. of cylinder) – (T.S.A. of cones)

→ 1100 – 713.62

→ 386.76 cm²

Total surface area of the remaining solid = 386.76 cm²