Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 1 • Question 17

The shapes K and L are two similar cylinders. The height of cylinder K is three times the height of cylinder L. Jacob says that the area of the circular face of cylinder K must be three times the area of the circular face of cylinder L. Is Jacob correct? Tick the correct box. Give a reason for your answer.

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Video Transcript

The shapes K and L are two similar cylinders. The height of cylinder K is three times the height of cylinder L. Jacob says that the area of the circular face of cylinder K must be three times the area of the circular face of cylinder L. Is Jacob correct? Tick the correct box. Give a reason for your answer.

As the two shapes are similar, K must be a scale factor enlargement of L. This means that each length of K will be three times each of the lengths in cylinder L. If the height of cylinder L is ℎ, then the height of cylinder K is three ℎ. This means that our length scale factor is equal to three.

We therefore know that if the radius of the circular face of cylinder L is 𝑟, then the radius of the circular face of cylinder K will be three 𝑟. Jacob says that the area of the circular face of cylinder K is three times the area of the circular face of cylinder L. However, we know that the area scale factor is equal to the length scale factor squared. In this case, as the length scale factor is three, the area scale factor will be equal to three squared. This is equal to nine. Therefore, the area of the circular face in cylinder K will be nine times the area of the circular face in cylinder L.

We can check this by using the formula for the area of a circle, which is equal to 𝜋𝑟 squared or 𝜋 multiplied by the radius squared. The area of the circular face of cylinder L would just be 𝜋𝑟 squared, as the radius is equal to 𝑟. The area of the circular face of cylinder K would be equal to 𝜋 multiplied by three 𝑟 squared, as three 𝑟 is the radius. Three 𝑟 multiplied by three 𝑟 is equal to nine 𝑟 squared. This means that the area of the circular face of cylinder K is equal to nine 𝜋𝑟 squared.

We can see once again that the area of the circular face of cylinder K is nine times the circular face of cylinder L. Our answer is no, Jacob is not correct as the area will be nine times bigger, not three times bigger.