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.
Practise Edexcel A Level Maths 11.11 Modelling with Differential Equations with exam-style questions for A Level Maths. 51 questions, matched to the Edexcel A Level Maths (9MA0) 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.
