The diagram shows a cube with edges of length $x$ cm and a sphere of radius 3 cm - Edexcel - GCSE Maths - Question 10 - 2021 - Paper 1

The surface area (SA) of a sphere is given by the formula: $SA_{sphere} = 4\pi r^2,$ where $r$ is the radius. Substituting $r = 3$ cm: $SA_{sphere} = 4\pi (3^2) = 36\pi.$

We need to set the surface area of the cube equal to the surface area of the sphere: $6x^2 = 36\pi.$

Dividing both sides by 6 gives: $x^2 = 6\pi.$ Therefore, $x = \sqrt{6\pi}.$ To express in the form $x = \sqrt{kT}$, where $k$ is an integer, note that: $\pi$ is approximately 3.14, thus: $k = 6\cdot 3 = 18$ and $T = \pi$, which leads to: $x = \sqrt{6\pi} = \sqrt{18\cdot \frac{\pi}{3}}.$ Hence, we can rewrite it as: $x = \sqrt{kT}$ where $k = 18$ is an integer.