Video: CBSE Class X • Pack 3 • 2016 • Question 18

CBSE Class X • Pack 3 • 2016 • Question 18


Video Transcript

A sphere of diameter 12 centimetres is dropped into a right circular cylindrical vessel partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by three and five-ninths of a centimetre. Find the diameter of the cylindrical vessel.

Let’s begin by calculating the volume of the sphere. The volume of a sphere with a radius 𝑟 is given by the formula four-thirds multiplied by 𝜋 multiplied by 𝑟 cubed. Since the sphere has a diameter of 12 centimetres, its radius can be found by halving 12. Half of 12 is six centimetres. Substituting what we know into the formula, we get that the volume of our sphere is four-thirds multiplied by 𝜋 multiplied by six cubed. Six cubed is six multiplied by six multiplied by six. And we can divide six by three to get two.

That leaves us with four times two times six times six multiplied by 𝜋. Six multiplied by six is 36. We can then multiply 36 by four by doubling it once and then doubling it again. 36 doubled is 72. Timesing it by two again, and we get 144. Finally, we can multiply this by two to get us 288. The volume of the sphere is 288𝜋 centimetres cubed.

The formula for volume of a cylinder is 𝜋𝑟 squared ℎ, where 𝑟 is the radius of the cylinder and ℎ is its height. We don’t know the volume. But we do know that the change in its height is three and five-ninths of a centimetre. The displacement represents the volume of the sphere. So we can substitute three and five-ninths into the formula for volume of the cylinder and equate it to 288𝜋 which we calculated as being the volume of the sphere.

Before we do any further calculations, we should convert three and five-ninths into an improper fraction. To do that, we first multiply three by nine which tells us the total number of ninths in three wholes. Three multiplied by nine is 27. So we have twenty-seven ninths and an additional five-ninths. That gives us a total of thirty-two ninths. So our equation becomes 288𝜋 equals 𝜋 times radius squared times thirty-two ninths. We can divide through by 𝜋. And we get 288 equals thirty-two ninths multiplied by 𝑟 squared.

Next, we can divide through by 32. 32 is a factor of 288. But if we haven’t spotted that, we could’ve repeatedly divided both sides of this equation by two. In fact, 288 divided by 32 is nine. Notice, 32 multiplied by 10 is 320. 288 is one less 32 than this. So we know that nine is equal to one-ninth of 𝑟 squared. Next, we’ll multiply both sides of this equation by nine, which gives us 𝑟 squared is equal to 81.

To find the value of 𝑟, we’ll square root everything. The square root of 81 is nine. And we should always consider that there’s actually two solutions whenever we square root. We get a positive and a negative. However, the radius represents a length. So it must be a positive value since the length can never be negative. We calculated the radius of the vessel to be nine centimetres. The diameter is double the radius. Nine multiplied by two is 18. So the diameter is 18 centimetres.

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