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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.