A cylindrical ice pillar is melting in a laboratory. At time ttt minutes, the pillar has a radius r mmr\text{ mm}r mm and a length L=8r mmL = 8r\text{ mm}L=8r mm.
The cross-sectional area of the pillar, AAA, is decreasing at a constant rate of 1.2 mm2 min−11.2\text{ mm}^2\text{ min}^{-1}1.2 mm2 min−1.
Find the value of drdt\frac{dr}{dt}dtdr when r=3r = 3r=3.
Determine the rate of decrease of the volume of the pillar when r=5r = 5r=5.
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.