A spherical drop of industrial lubricant is being injected into a precision-engineered cavity. The volume, VVV, of the drop is increasing at a constant rate of 180π mm3 s−1180\pi \text{ mm}^3\text{ s}^{-1}180π mm3 s−1. Calculate the rate of increase of the radius, rrr, of the drop in mm s−1 \text{mm s}^{-1}mm s−1 at the moment when the radius is exactly 3 mm3 \text{ mm}3 mm. [The volume VVV of a sphere of radius rrr is given by V=43πr3V = \frac{4}{3}\pi r^3V=34πr3]
The depth of sediment, y metresy \text{ metres}y metres, settling at the bottom of an industrial filtration tank is monitored. The rate of change of the depth of the sediment is modeled by the differential equation
dydt=ky2 \frac{\text{d}y}{\text{d}t} = \frac{k}{y^2} dtdy=y2kwhere kkk is a positive constant and ttt hours is the time after monitoring began. Given that:
Solve the differential equation to determine the value of TTT. Give your answer to one decimal place.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
