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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.