Question

ANSWER CORRECTLY AND GET BRAINLIEST OR YOUR ACCOUNT WILL BE DELETED. A solid cone of radius 5cm and height 12 cm is melted and recast into 150 hemispheres of radius 1 cm. Find the increase in surface area on recasting in terms of pi

Answer 1

Answer

360 π cm²

Solution

Given

A solid cone of radius = 5 cm

Height = 12 cm

Radius of 150 hemispheres = 1 cm

Since it's melted and recasted, the volume of cone would be equal to the volume of 150 hemispheres altogether.

We know that

surface area of cone = πr(l + R) (R be the radius of cone and l be the slant height)

l = √(h² + R²)

→ l = √(12² + 5²)

→ l = √(144 + 25)

→ l = √169

→ l = 13 cm

So, surface area of cone = 5π(13 + 5)

→ 90π cm²

And, surface area of hemispheres = 3πr² (r be the radius of hemispheres)

→ surface area of 150 hemispheres = 150 × 3πr²

→ 150 × 3π × 1²

→ 450π cm²

So, increase in surface area = 450π - 90π = 360π cm²

Answer 2

$\huge\underline\mathfrak{Question-}$

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A solid cone of radius 5cm and height 12 cm is melted and recast into 150 hemispheres of radius 1 cm. Find the increase in surface area on recasting in terms of π.

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$\huge\underline\mathfrak{Answer-}$

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$\large{\underline{\boxed{\rm{\blue{Increase\:in\:surface\:area\:=\:360\pi}}}}}$

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$\huge\underline\mathfrak{Explanation-}$

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$\begin{lgathered}\bold{Given} \begin{cases}\sf{Radius\:of\:each\:hemisphere\:=\:1\:cm} \\ \sf{Radius\:of\:cone\:=\:5\:cm}\\ \sf{Height\:of\:cone\:=\:12\:cm}\end{cases}\end{lgathered}$

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To find :

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• Increase in surface area on recasting 150 hemispheres from a solid cone in terms of π.

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Solution :

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In these type of questions, we have to equate the volume of both of the solids.

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We know,

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$\large{\boxed{\rm{Surface\:Area\:of\:cone\:=\:\pi\:r(r+l)}}}$

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$\rule{200}2$

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We have to calculate the slant height ( l ) first.

l² = r² + h²

$\implies$ l² = 5² + 12²

$\implies$ l² = 25 + 144

$\implies$ l² = 169

$\implies$ l = √169

$\implies$ l = 13 cm

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$\rule{200}2$

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Now, putting the values in above formula,

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$\implies$ Surface area of cone = π (5) ( 5 + 13 )

$\implies$ Surface area of cone = 5π × 18

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$\therefore$ Surface area of cone = 90π cm²

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We also know that,

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$\large{\boxed{\rm{Surface\:Area\:of\:hemisphere\:=\:3\pi\:r^2}}}$

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Putting the values,

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$\implies$ Surface area of hemisphere = 3π (1)²

$\implies$ Surface area of hemisphere = 3π

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It is given that cone is melting to recast 150 hemispheres. So we have to multiply the surface area of hemispheres by 150 to find the surface area of 150 such hemispheres.

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$\implies$ Surface area of 150 hemispheres = 150 × surface area of 1 hemisphere

$\implies$ Surface area of 150 hemispheres = 150 × 3π

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$\therefore$ Surface area of 150 such hemispheres = 450π cm²

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Increase in surface area = surface area of hemisphere - surface area of cone

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Putting the values,

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$\implies$ Increase in surface area = 450π - 90π

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$\large{\underline{\boxed{\rm{\blue{\therefore\:Increase\:in\:surface\:area\:=\:360\pi}}}}}$