Metallic spheres of radii 6 cm , 8 cm and 10 cm respectively are melted to form a single solid sphere. Find the radius of the resulting sphere. Also find the surface area of the new sphere.

Given: The radius ( r 1 ) of 1 s t metallic sphere = 6 cm The radius ( r 2 ) of 2 n d metallic sphere = 8 cm The radius ( r 3 ) of 3 r d metallic sphere = 10 cm Let the radius of the resulting sphere be r . When we convert one solid to another, the volume remain the same The volume of the three spheres = Volume of the resulting sphere ⇒ 4 3 π r 1 3 + r 2 3 + r 3 3 = 4 3 πr 3 ⇒ 4 3 π 6 3 + 8 3 + 10 3 = 4 3 πr 3 ⇒ r 3 = 216 + 5 12 + 1000 = 1728 cm 3 ⇒ r = 12 cm Therefore, the radius of the resulting sphere so formed is 12 c m . We know that, the surface area of the new sphere = 4 π r 2 The surface area of the new sphere = 4 × 22 7 × 12 × 12 = 88 × 144 7 = 1810 . 3 cm 2 Hence, the surface area of the new sphere is 1810 . 3 cm 2

A hemispherical tank full of water is emptied by a pipe at the rate of 3 4 7 litres per second. How much time will it take to empty half the tank , if it is 3 m in diameter?

Radius of base of tank = D 2 = 3 2 m Volume of tank = 2 3 πr 3 = 2 3 × 22 7 × 3 2 3 = 99 14 m 3 = 99 14 × 1000 l i t r e s = 99000 14 l i t r e s Volume of water to be emptied = 1 2 × volume of tank = 1 2 × 99000 14 l i t r e s = 99000 28 l i t r e s Time taken to empty 3 4 7 = 25 7 l i t r e s = 1 s e c Time taken to empty 1 l i t r e s = 7 25 s e c Time taken to empty 99000 28 l i t r e s = 99000 28 × 7 25 = 69300 700 = 990 s e c = 990 60 m i n = 16 . 5 m i n Time taken to empty half tank is 16 minute and 5 second.

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 13 - Surface Areas and Volumes>Assam Board-2020>Q 7

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$2$ cubes each of volume $64 {cm}^{3}$ are joined end to end. The surface area of the resulting cuboid is :

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 13 - Surface Areas and Volumes>Assam Board-2020>Q 8

A toy is in the form of a cone of radius $3.5 cm$ mounted on a hemisphere of same radius. The total height of the toy is $15.5 cm$ . Find the total surface area of the toy.