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.
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.