A spherical raindrop increases in volume, VVV, at a constant rate of 450π mm3 min−1450\pi \text{ mm}^3\text{ min}^{-1}450π mm3 min−1 as it falls through a mist. Calculate the rate at which the radius, rrr, is increasing in mm min−1\text{mm min}^{-1}mm min−1 at the moment when r=15 mmr = 15 \text{ mm}r=15 mm. [The volume VVV of a sphere of radius rrr is given by the formula V=43πr3V = \frac{4}{3}\pi r^3V=34πr3]
The mass, M gramsM \text{ grams}M grams, of a crystal growing in a saturated solution is monitored over time. The rate of increase in the mass of the crystal is modeled by the differential equation
dMdt=kM \frac{\text{d}M}{\text{d}t} = \frac{k}{\sqrt{M}} dtdM=Mkwhere kkk is a positive constant and ttt hours is the time after monitoring began. Given that:
Solve the differential equation to find 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.