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1.9 Calculus

1.9 Calculus

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Question 625

A biologist is studying the population, PPP, of a specific strain of bacteria in a petri dish. The rate of change of the population is modeled by the differential equation

dPdt=3P(4−t)8 \frac{dP}{dt} = \frac{3P(4 - t)}{8} dtdP​=83P(4−t)​

where t≥0t \ge 0t≥0 is the time in hours since the start of the experiment. Initially, the population is 40 units.

a.

Solve the differential equation to show that the population at time ttt is given by

P=40e316(8t−t2)for 0<t<c P = 40 e^{\frac{3}{16}(8t - t^2)} \quad \text{for } 0 < t < c P=40e163​(8t−t2)for 0<t<c

where ccc is a constant to be found that represents the time when the population first returns to its initial value.

[5]
b.

Find the exact maximum population predicted by this model. Fully justify that your answer is a maximum.

[5]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

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.

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