An industrial spill of a viscous lubricant is spreading across a flat floor. The area of the spill, L m2L \text{ m}^2L m2, at time ttt hours after the spill is discovered, is modelled by the differential equation
dLdt=LL5t2,t>0 \frac{dL}{dt} = \frac{L\sqrt{L}}{5t^2}, \quad t > 0 dtdL=5t2LL,t>0Given that the spill covers 16 m216 \text{ m}^216 m2 at the moment it is discovered (t=1t = 1t=1),
show that
L=(ptqt+r)2 L = \left( \frac{pt}{qt + r} \right)^2 L=(qt+rpt)2where ppp, qqq, and rrr are integers to be found.
According to the model, find the limiting value of the area covered by the spill as t→∞t \to \inftyt→∞.
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.