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.
717 exam-style questions on Edexcel A Level Maths Differentiation, covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change. Each one has a worked solution and a mark scheme showing where the marks go.