Centre, radius and volume of a sphere from the ends of a diameter, and minimising the surface area of a cuboid box

In this question, all lengths are measured in metres.

The points \(P(-1,4,7)\) and \(Q(5,-2,3)\) are the ends of a diameter \([PQ]\) of a sphere.

1. Find
   1. the centre of the sphere, giving its coordinates;
   2. the length of its radius.[4]
2. Calculate its volume.[2]

A closed storage box is a cuboid with base length \(3x\), base width \(x\) and height \(y\). The volume of the box is \(72\text{ m}^3\) and its total surface area is \(A\text{ m}^2\).

3. 1. Write down an equation connecting \(x\) and \(y\) for the volume of the box.
   2. Show that \(A=6x^2+\dfrac{192}{x}\).[4]

The value of \(A\) is least when \(x=k\).

4. 1. Find the least value of \(A\).
   2. Find \(\dfrac{\mathrm{d}^2A}{\mathrm{d}x^2}\) when \(x=k\).
   3. Justify, using the result of part (d)(ii), that \(A\) is least when \(x=k\).[5]