A laboratory-grown synthetic crystal takes the form of a regular tetrahedron, which consists of 4 congruent equilateral triangular faces meeting at 4 vertices. The side length of the tetrahedron is denoted by L mmL\text{ mm}L mm.
Show that the total surface area, S mm2S\text{ mm}^2S mm2, of the tetrahedron is given by
S=3L2 S = \sqrt{3}L^2 S=3L2Given that the volume, V mm3V\text{ mm}^3V mm3, of the tetrahedron is given by
V=L362 V = \frac{L^3}{6\sqrt{2}} V=62L3show that dVdS=624L\dfrac{\text{d}V}{\text{d}S} = \dfrac{\sqrt{6}}{24}LdSdV=246L.
The surface area of the crystal is increasing at a constant rate of 0.14 mm2 s−10.14\text{ mm}^2\text{ s}^{-1}0.14 mm2 s−1.
Find the rate of change of the volume of the crystal when L=6L = 6L=6, giving your answer to 2 significant figures.
