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.
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.