Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 3 • Question 12

A cylindrical tube is filled with four tennis balls. The tennis balls are tightly packed inside the tube such that they touch both faces of the cylinder as shown in the diagram. (a) The radius of a tennis ball is 6.8 cm. Work out the volume of one tennis ball. (b) Calculate the volume of the cylinder.

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Video Transcript

A cylindrical tube is filled with four tennis balls. The tennis balls are tightly packed inside the tube such that they touch both faces of the cylinder as shown in the diagram. Part a) The radius of a tennis ball is 6.8 centimetres. Work out the volume of one tennis ball.

There is also a part b that we’ll come onto. And we’re also told that the volume of a sphere is equal to four-thirds 𝜋𝑟 cubed. And to solve part a, we’re gonna utilize this formula. And we know we can use this formula because first of all we’re trying to find the volume of one tennis ball. And a tennis ball is a sphere. And also, we’re told the radius of that tennis ball and the radius is 6.8 centimetres. So what we can do is substitute in our value for 𝑟 — so that’s 6.8 — into our formula for the volume of a sphere.

So we can say that the volume of the tennis ball — so 𝑉 T — is equal to four-thirds multiplied by 𝜋 then multiplied by 6.8 cubed. And this will give us 1317.089682 centimetres cubed. What we’re gonna do though is round this to a sensible degree of accuracy. So when I do that, I get 1317.09 centimetres cubed and that’s to two decimal places. I decided to do it to two decimal places.

And the way I did that was I looked to the second decimal place and that was eight. Then I looked to the number after it. So I’ve drawn a orange line underneath that. That’s the nine and that’s our deciding number. Because this is five or above, we then round the eight to a nine. So we get 1317.09 centimetres cubed to two decimal places. So that’s part a complete.

So for part b, what we’re asked to do is calculate the volume of the cylinder. And how we’re gonna do this? Well, first of all, we remind ourselves that we we’re told the radius in part a. And the radius was equal to 6.8 centimetres. So therefore, we can say that the diameter of the tennis ball is gonna be equal to 13.6 centimetres. And we know that because the diameter is equal to two multiplied by the radius cause it’s twice the radius.

And the reason we want to find the diameter of the sphere is cause the diameter of the sphere is also the height of the sphere. And we want that because eventually we want to find the height of the cylinder. And that’s because we have a formula for the volume of a cylinder. And the volume of a cylinder is equal to 𝜋𝑟 squared multiplied by the height. So therefore, we can work out the height of the cylinder by multiplying 13.6 by four. And that’s because the height of one sphere or one tennis ball was 13.6 cause that was the diameter. So when we do that, we get 54.4.

And the reason we know we can do that to work out the height of the cylinder is because we’re told that the tennis balls are tightly packed inside the tube so they touch both faces. So therefore, we can use this value to substitute in to our formula for the volume of a cylinder. And as we said, the volume of a cylinder is equal to 𝜋𝑟 squared ℎ.

So therefore, it’s gonna be equal to 𝜋 multiplied by 6.8 squared multiplied by 54.4. And that’s because 6.8 is the radius and 54.4 is the height, which will give us a volume of 7902.53809, which again if I round to two decimal places, it’s gonna give us a volume of the cylinder of 7902.54 centimetres cubed. And I got that again because I looked at the second decimal place which was three then the deciding number or the deciding digit after it; it’s an eight. This is five or above. So we round up to a four.

So therefore, we can say the answer to part a is the volume of a tennis ball is 1317.09 centimetres cubed to two decimal places and for part b the volume of the cylinder is 7902.54 centimetres cubed to two decimal places.

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