A crystal in the form of a regular tetrahedron, with side length s cms\text{ cm}s cm, is growing in a laboratory. The tetrahedron consists of 4 congruent equilateral triangular faces.
Show that the total surface area, S cm2S\text{ cm}^2S cm2, of the tetrahedron is given by
S=3s2 S = \sqrt{3}s^2 S=3s2Given that the volume, V cm3V\text{ cm}^3V cm3, of the tetrahedron is given by
V=212s3 V = \frac{\sqrt{2}}{12}s^3 V=122s3Prove that dVdS=624s\dfrac{dV}{dS} = \dfrac{\sqrt{6}}{24}sdSdV=246s.
The surface area of the crystal is increasing at a constant rate of 0.12 cm2 s−10.12\text{ cm}^2\text{ s}^{-1}0.12 cm2 s−1.
Determine the rate of change of the volume of the crystal at the instant when s=8s = 8s=8, giving your answer to 2 significant figures.
