Skip to content

Course home

Sign up

1.7 Differentiation

EasyMediumHard
1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495
Question 61

The concentration CCC mg/L of a specific industrial pollutant in a processing tank ttt hours after a chemical filtration process begins is modelled by the equation

C=C0e−kt C = C_0 e^{-kt} C=C0​e−kt

where C0C_0C0​ is the initial concentration and kkk is a positive constant. The model is designed to represent the efficiency of the filtration over a long period.

a.

It takes 10 hours for the concentration of the pollutant to reduce to 50%50\%50% of its initial value.

Determine the number of days required for the concentration to be reduced by at least 99%99\%99% from its initial value. Give your answer to one decimal place.

[5]
b.

Determine the percentage of the initial concentration remaining in the tank after 6 days. Give your answer to two significant figures.

[3]
c.

Explain why this model only provides an estimate for the actual concentration of the pollutant in the tank.

[1]
d.

Explain why this model is physically unrealistic for the tank as t→∞t \to \inftyt→∞.

[1]
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.

Question bank