The diagram shows a cylinder and a cone. The cylinder has radius 2 cm and height 9 cm. The cone has radius r cm and height h cm. The ratio r : h is 1 : 4. The volume of the cone is equal to the volume of the cylinder. Work out the value of r. [The volume V of a cone with radius r and height h is V = \frac{1}{3} \pi r^{2} h.]

GCSE OCR Maths Volume: The diagram shows a cylinder and

The diagram shows a cylinder and a cone - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

Question 14

The diagram shows a cylinder and a cone. The cylinder has radius 2 cm and height 9 cm. The cone has radius r cm and height h cm. The ratio r : h is 1 : 4. The volu... show full transcript

Worked Solution & Example Answer:The diagram shows a cylinder and a cone - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

Step 1

Calculate the volume of the cylinder

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Answer

The volume V of the cylinder can be calculated using the formula:

$V = \pi r^{2} h$

For the cylinder:

Radius = 2 cm
Height = 9 cm

Thus,

$V_{cylinder} = \pi (2)^{2} (9) = 36\pi \text{ cm}^{3}$

Step 2

Set the volumes equal

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Answer

Given that the volume of the cone is equal to the volume of the cylinder, we have:

$V_{cone} = V_{cylinder}$

From the cone's volume formula:

$V_{cone} = \frac{1}{3} \pi r^{2} h$

Setting these equal, we have:

$\frac{1}{3} \pi r^{2} h = 36\pi$

Step 3

Substituting the ratio of r and h

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Answer

From the ratio given, ( r : h = 1 : 4 ), we can express h as:

$h = 4r$

Substituting this into the volume equation gives:

$\frac{1}{3} \pi r^{2} (4r) = 36\pi$

Step 4

Solve for r

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Answer

Simplifying the equation:

$\frac{4}{3} \pi r^{3} = 36\pi$

Dividing both sides by (\pi):

$\frac{4}{3} r^{3} = 36$

Multiplying through by 3:

$4r^{3} = 108$

Now divide by 4:

$r^{3} = 27$

Taking the cube root of both sides yields:

$r = 3 \text{ cm}$

The diagram shows a cylinder and a cone - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

Worked Solution & Example Answer:The diagram shows a cylinder and a cone - OCR - GCSE Maths - Question 14 - 2018 - Paper 1

Calculate the volume of the cylinder

Set the volumes equal

Substituting the ratio of r and h

Solve for r

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