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.
864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.