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1.7 Differentiation

1.7 Differentiation

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Question 13

An industrial desalination plant monitors the mass of salt extracted from seawater. The process is modeled by two variables:

  • ttt, the number of hours the plant has been operating since its daily maintenance cycle at 08:00.
  • SSS, the total mass of salt, in kg, extracted since 08:00.

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.

a.

Show that SdSdt=400(10−t)\displaystyle S \frac{dS}{dt} = 400(10 - t)SdtdS​=400(10−t).

[3]
b.

Hence, show that S2=400t(20−t)S^2 = 400t(20 - t)S2=400t(20−t).

[4]
c.

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).

[5]
Markscheme

1.7 Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.7 Differentiation

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.

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