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

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:

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

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

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

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:

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

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:

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