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

The volume of the square-based pyramid can be calculated first. The area of the base is:

$\text{Area} = \text{side}^2 = 15^{2} = 225 \, \text{cm}^{2}$

Let the perpendicular height be $h$. Thus, the volume of the pyramid is:

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

The volume of the hemisphere with radius 6 cm is given by:

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

Approximately, this is $144\pi \approx 452.39 \, \text{cm}^3$.

If the volume of the glass is reduced by 30%, it means 70% of the original volume remains:

$0.7 \times V_{\text{pyramid}} = V_{\text{pyramid}} - V_{\text{hemisphere}}$

Substituting the volumes we have:

$0.7(75h) = 75h - 144\pi$

Simplifying gives us:

$52.5h = 75h - 144\pi$

Rearranging this, we find:

$75h - 52.5h = 144\pi$

$22.5h = 144\pi$

Thus, we can solve for $h$:

$h = \frac{144\pi}{22.5} = 6.4\pi \approx 20.1 \, \text{cm}$