The volume, VVV, of a spherical weather balloon is increasing at a constant rate of 120π cm3 s−1120\pi \text{ cm}^3\text{ s}^{-1}120π cm3 s−1. Find the rate of increase of the radius, rrr, of the balloon in cm s−1\text{cm s}^{-1}cm s−1 at the instant when the radius is 6 cm6 \text{ cm}6 cm. [The volume VVV of a sphere of radius rrr is given by the formula V=43πr3V = \frac{4}{3}\pi r^3V=34πr3]
The height of a pile of sand, h metresh \text{ metres}h metres, under a conveyor belt is monitored over time. The rate of increase in the height of the pile is modeled by the differential equation
dhdt=kh2 \frac{\text{d}h}{\text{d}t} = \frac{k}{h^2} dtdh=h2kwhere kkk is a positive constant and ttt hours is the time after the measurement began. Given that:
Solve the differential equation to find 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.
