A research scientist is monitoring the internal pressure PPP (in kPa) of a sealed bioreactor during a temperature-controlled experiment. The time ttt, in hours, is measured from the start of the experiment. The scientist models the rate of change of pressure as being directly proportional to 15−tP\displaystyle \frac{15 - t}{P}P15−t.
After 5 hours, the pressure is 250 kPa and the rate of increase of pressure is 20 kPa per hour.
Show that PdPdt=500(15−t)\displaystyle P \frac{\text{d}P}{\text{d}t} = 500(15 - t)PdtdP=500(15−t).
Hence, show that P2=500t(30−t)P^2 = 500t(30 - t)P2=500t(30−t).
The experiment began at 06.00. (i) The researcher stops monitoring the bioreactor when the rate of pressure change drops below 10 kPa per hour. Using the results in parts (a) and (b), determine the earliest time that the researcher stops monitoring. (ii) Explain why the model used by the scientist is not valid at 06.00.
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.