2025 DHS Promo Q10

[It is given that a sphere of radius \(R\) has surface area \(4\pi R^2\) and volume \(\frac{4}{3}\pi R^3\), and a cone of radius \(r\) and height \(h\) has volume \(\frac{1}{3}\pi r^2h\).]

A company sells decorative balloons placed inside a conical gift box. Each balloon is in the shape of a sphere of fixed radius 5 cm, placed inside a right circular cone with radius \(r\) cm and height \(h\) cm. The balloon touches the sides of the cone and its bottom just reaches the cone’s base. (See diagram below). Assume the cone is made of a thin material and has negligible thickness.

1. Show that \(r = \frac{5h}{\sqrt{h^2-10h}}\).[2]
2. For cost efficiency, the gift box should not be unnecessarily large.\nTo minimise empty space, the volume of the cone should be as small as possible.\nUsing differentiation, find the exact minimum volume of the cone.[5]
3. Once the balloon is exposed to sunlight, it starts to deflate but remains spherical.\nThe surface area of the balloon is decreasing at a constant rate of \(\frac{1}{2}\pi\text{ cm}^2/\text{min}\).\nFind the rate of decrease of the volume of the balloon 8 minutes after it was exposed.[4]