An industrial desalination plant monitors the mass of salt extracted from seawater. The process is modeled by two variables:
The engineers model the rate of salt extraction as being directly proportional to 10−tS\displaystyle \frac{10 - t}{S}S10−t.
After 2 hours of operation, the rate of salt extraction is 803\displaystyle \frac{80}{3}380 kg per hour and the total mass of salt extracted is 120 kg.
Show that SdSdt=400(10−t)\displaystyle S \frac{dS}{dt} = 400(10 - t)SdtdS=400(10−t).
Hence, show that S2=400t(20−t)S^2 = 400t(20 - t)S2=400t(20−t).
The plant began its operation at 08:00. (i) The plant's efficiency drops over time. The operation is paused for a second maintenance cycle when the rate of salt extraction falls below 15 kg per hour. Using the results in parts (a) and (b), calculate the time of day when the operation is paused. (ii) Explain why the model used by the engineers is not valid at the start of the process (08:00).
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.