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.
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.