Question

A solid cone of radius 3 cm and slant height 5 cm is cut by a plane through the vertex A into two equal pieces. What is the total surface area of a piece?

Answer 1

$\pi rl/2 + \pi r^{2}/2 +d( l^{2} - d^{2}/4)/2$
if its divided through the top vertex A.
d is the diametre.
then ur cone will have 3 planes....curved,triangular(side),semi circular(bottom).

answer= 23.56+14.13+48=85.69.

Answer 2

$\bf Given \begin{cases} & \sf{Radius,\; r = \bf{3\;cm}} \\ & \sf{Height, \;h = \bf{5\;cm}} \end{cases}$

To find: Total surface area of a piece?

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$\underline{\sf{\bigstar\; According\;to\;the\;Question\;:}}$

A solid cone is cut by a plane through the vertex A into two equal pieces.

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$\setlength{\unitlength}{1cm}\begin{picture}(6, 4)\linethickness{0.26mm} \qbezier(5.8,2.0)(5.8,2.3728)(4.9799,2.6364)\qbezier(4.9799,2.6364)(4.1598,2.9)(3.0,2.9)\qbezier(3.0,2.9)(1.8402,2.9)(1.0201,2.6364)\qbezier(1.0201,2.6364)(0.2,2.3728)(0.2,2.0)\qbezier(0.2,2.0)(0.2,1.6272)(1.0201,1.3636)\qbezier(1.0201,1.3636)(1.8402,1.1)(3.0,1.1)\qbezier(3.0,1.1)(4.1598,1.1)(4.9799,1.3636)\qbezier(4.9799,1.3636)(5.8,1.6272)(5.8,2.0)\put(0.2,2){\line(1,0){2.8}}\put(4.9,4){\sf{5 cm}}\put(1.4,1.6){\sf{3 cm}}\qbezier(.185,2.05)(.7,3)(3,6.5)\qbezier(5.8,2.05)(5.3,3)(3,6.5)\put(3,2.02){\circle*{0.15}}\put(2.7,2){\dashbox{0.01}(.3,.3)}\multiput(3,1.2)(0,0.5){11}{\line(0,1){0.2}}\end{picture}$

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Now,

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We know that,

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$\star\;{\boxed{\sf{\purple{TSA_{\;(cone)} = \pi r(r + l)}}}}$

Putting values,

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$:\implies\sf \dfrac{22}{7} \times 3 \bigg(3 + 5 \bigg)$

$:\implies\sf \dfrac{66}{7} \bigg(3 + 5 \bigg)$

$:\implies\sf \dfrac{66}{7} \times 8$

$:\implies{\boxed{\frak{\pink{75.42\;cm^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{The\;total\;surface\;area\;of\;cone\;is\; \bf{75.42\;cm^2}.}}}$

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It is divided into two pieces.

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Therefore,

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Total surface area of a single piece is,

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$:\implies\sf \dfrac{75.42}{2}$

$:\implies{\boxed{\frak{\pink{37.71\;cm^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;The\;total\;surface\;area\;of\;a\;piece\;cone\;is\; \bf{37.71\;cm^2}.}}}$