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Surface Areas and Volumes

Short Answer Type

351. Three solid spheres of iron whose diameters are 2 cm, 12 cm and 16 cm respectively are melted into a single solid sphere. Find the radius of the solid sphere.

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352. A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1 : 2 : 3.

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353. A spherical ball of lead 3 cm in radius is melted and recast into three spherical balls. If the radii of two balls are

3 over 2

cm and 2 cm, find the diameter of the third ball.

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354. The volumes of two spheres are in the ratio 64 : 27. Find the difference of their surface areas, if the sum of their radii is 7 cm.

Let the radii of the two spheres be r¹ cm and r₂ cm respectively. Then,

straight v subscript 1 over straight v subscript 2 equals fraction numerator begin display style 4 over 3 end style πr subscript 1 cubed over denominator begin display style 4 over 3 end style πr subscript 2 cubed end fraction equals 64 over 27 rightwards double arrow space space open parentheses r subscript 1 over r subscript 2 close parentheses cubed equals open parentheses 4 over 3 close parentheses cubed rightwards double arrow space space space r subscript 1 over r subscript 2 equals 4 over 3 space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis straight r subscript 1 plus straight r subscript 2 equals 7 space cm space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

From (1) and (2)

straight r subscript 1 space equals space 4 space cm straight r subscript 2 space equals space 3 space cm

therefore

Difference of their surface areas

_{$equals 4 πr subscript 1 squared minus 4 πr subscript 2 squared equals space 4 πr left parenthesis straight r subscript 1 squared minus straight r subscript 2 squared right parenthesis equals space 4 straight pi space left parenthesis 4 squared minus 3 squared right parenthesis equals space 4 straight pi left parenthesis 16 minus 9 right parenthesis equals 4.22 over 7.7 equals space 88 space cm squared$}

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355. A hemispherical bowl of internal diameter 36 cm contains a liquid. This liquid is to be filled in cylindrical bottles of radius 3 cm and height 6 cm. How many bottles are required to empty the bowl?

open square brackets Use space straight pi equals 22 over 7 close square brackets

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356. A hemispherical bowl is made of steel 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl.

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357.

A spherical iron shell with 8 cm external diameter weighs $1860 4 over 7 straight g.$
Find the sick ness of the shell if the density of metal is 12 g/cm³ .