The concentration, CCC mg/L, of a particular chemical residue in a large processing tank, ttt hours after a neutralising catalyst is introduced, is modelled by the differential equation
dCdt=−k(C−10)2 \frac{\text{d}C}{\text{d}t} = -k(C - 10)^2 dtdC=−k(C−10)2where kkk is a constant and C>10C > 10C>10.
Given that the concentration of the residue:
Solve the differential equation to show that, according to the model
C=at+bct+d C = \frac{at + b}{ct + d} C=ct+dat+bwhere a,b,ca, b, ca,b,c and ddd are integers to be found.
Hence find, according to the model, the time taken for the concentration of the residue to fall to 404040 mg/L. Give your answer to the nearest hour.
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.