Calculate the volume of a sphere with radius 6 cm. [The volume V of a sphere with radius r is V = \frac{4}{3} \pi r^3 ] (b) An ornament is made from a solid glass square-based pyramid. The base has side length 15 cm. A hemisphere with radius 6 cm is cut out of the base of the pyramid. This reduces the volume of glass contained in the ornament by 30%. Calculate the perpendicular height of the pyramid. [The volume of a pyramid is \frac{1}{3} \times \text{area of base} \times \text{perpendicular height}. A hemisphere is half a sphere.]

GCSE OCR Maths Pythagoras: Calculate the volume of a sphere

Calculate the volume of a sphere with radius 6 cm - OCR - GCSE Maths - Question 13 - 2018 - Paper 1

Question 13

Calculate the volume of a sphere with radius 6 cm. [The volume V of a sphere with radius r is V = \frac{4}{3} \pi r^3 ] (b) An ornament is made from a solid glass ... show full transcript

Worked Solution & Example Answer:Calculate the volume of a sphere with radius 6 cm - OCR - GCSE Maths - Question 13 - 2018 - Paper 1

Step 1

Calculate the volume of the sphere

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Answer

The volume V of a sphere can be calculated using the formula:

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

For a sphere with radius ( r = 6 ) cm:

$V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \; \text{cm}^3$

Step 2

Calculate the volume of the pyramid

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Answer

To find the volume of the pyramid, we first compute the area of the base, which is a square with side length 15 cm:

$\text{Area of base} = 15 \times 15 = 225 \; \text{cm}^2$

Let h be the height of the pyramid. The volume of the pyramid is given by:

$V_{pyramid} = \frac{1}{3} \times 225 \times h = 75h \; \text{cm}^3$

Step 3

Calculate the volume of the hemisphere

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Answer

The volume V of the hemisphere with radius 6 cm is:

$V_{hemisphere} = \frac{1}{2} \times \frac{4}{3} \pi (6)^3 = \frac{2}{3} \pi (216) = 144 \pi \; \text{cm}^3$

Step 4

Calculate the total volume reduction

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Answer

The total volume of glass contained in the ornament before the hemisphere is removed:

$V_{after} = V_{pyramid} - 0.3 \times V_{pyramid} = 0.7 \times V_{pyramid}$

Set this equal to the pyramid volume minus the hemisphere volume:

$0.7 \times 75h = 75h - 144 \pi$

Step 5

Solve for the height h

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Answer

Now, substituting the volume of the pyramid and rearranging gives:

$0.7 \times 75h = 75h - 144 \pi$

$52.5h = 75h - 144 \pi$

Combining terms:

$75h - 52.5h = 144 \pi$

$22.5h = 144 \pi$

Finally, solving for h:

$h = \frac{144 \pi}{22.5} \approx 20.1 \; \text{cm}$

Calculate the volume of a sphere with radius 6 cm - OCR - GCSE Maths - Question 13 - 2018 - Paper 1

Worked Solution & Example Answer:Calculate the volume of a sphere with radius 6 cm - OCR - GCSE Maths - Question 13 - 2018 - Paper 1

Calculate the volume of the sphere

Calculate the volume of the pyramid

Calculate the volume of the hemisphere

Calculate the total volume reduction

Solve for the height h

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