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 Proportion: 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 of a cylinder is given by the formula:

$V_{cylinder} = \pi r_{cylinder}^2 h_{cylinder}$

Here, the radius of the cylinder is 2 cm and the height is 9 cm. Therefore:

$V_{cylinder} = \pi (2^2)(9) = \pi (4)(9) = 36\pi$ \n\nThus, the volume of the cylinder is (36\pi) cm³.

Step 2

Express the Volume of the Cone

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Answer

The volume of a cone is given by the formula:

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

We need to express the height of the cone in terms of r. Given the ratio r : h = 1 : 4, we can say:

$h = 4r$ \n\nNow substituting this in the volume formula for the cone:

$V_{cone} = \frac{1}{3} \pi r^2 (4r) = \frac{4}{3} \pi r^3$ .

Step 3

Set Both Volumes Equal

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Answer

Since the volume of the cone is equal to the volume of the cylinder:

$\frac{4}{3} \pi r^3 = 36\pi$ \n\nWe can then cancel (\pi) from both sides:

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

Step 4

Solve for r

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Answer

To solve for r, first multiply both sides by 3:

$4r^3 = 108$ \n Next, divide both sides by 4:

$r^3 = 27$ \n Now take the cube root:

$r = 3$ cm.

Thus, the value of r is (3) 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

Express the Volume of the Cone

Set Both Volumes Equal

Solve for r

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