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The diagram shows a cylinder and a sphere - OCR - GCSE Maths - Question 6 - 2018 - Paper 4

Question 6

The diagram shows a cylinder and a sphere. The cylinder has radius 12 cm and height 30 cm. The cylinder and the sphere have the same volume. Work out the radius rc... show full transcript

Worked Solution & Example Answer:The diagram shows a cylinder and a sphere - OCR - GCSE Maths - Question 6 - 2018 - Paper 4

Step 1

Calculate the volume of the cylinder

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Answer

To find the volume of the cylinder, we use the formula:

$V_{cylinder} = \pi r^2 h$

Substituting the values:

Radius (r) = 12 cm
Height (h) = 30 cm

$V_{cylinder} = \pi (12)^2 (30)$ $V_{cylinder} = \pi \cdot 144 \cdot 30$ $V_{cylinder} = 4320\pi \text{ cm}^3$

Step 2

Set up the equation for the volume of the sphere

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Answer

Since the cylinder and the sphere have the same volume, we have:

$V_{sphere} = V_{cylinder}$

Using the formula for the volume of a sphere: $\frac{4}{3} \pi r_{cm}^3 = 4320\pi$

Step 3

Solve for the radius rcm of the sphere

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Answer

To solve for (r_{cm}), divide both sides by (\pi):

$\frac{4}{3} r_{cm}^3 = 4320$

Next, multiply both sides by (\frac{3}{4}):

$r_{cm}^3 = 4320 \cdot \frac{3}{4}$ $r_{cm}^3 = 3240$

Now take the cube root:

$r_{cm} = \sqrt[3]{3240}$ ( ext{Using a calculator, we find that } r_{cm} \approx 14.79 \text{ cm})$$

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