The concentration of a specific chemical, CCC mmol/L, in a bioreactor ttt hours after a growth process starts is modelled by the differential equation
dCdt=α−0.25C \frac{dC}{dt} = \alpha - 0.25C dtdC=α−0.25Cwhere α\alphaα is a positive constant. At the start of the process, there is no trace of the chemical in the reactor.
Solve the differential equation to show that C=4α(1−e−0.25t)C = 4\alpha(1 - e^{-0.25t})C=4α(1−e−0.25t).
In the long term, the concentration in the bioreactor stabilizes at 20 mmol/L.
Find the value of α\alphaα.
Determine the time, in hours, for the concentration to reach 15 mmol/L, 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.