The volume V m3V \text{ m}^3V m3 of a spherical weather balloon with radius r mr \text{ m}r m is given by the formula
V=43πr3 V = \frac{4}{3}\pi r^3 V=34πr3Find dVdr\frac{dV}{dr}drdV giving your answer in simplest form.
At time ttt seconds, helium is being pumped into the balloon such that the volume is increasing according to the differential equation
dVdt=1600(4t+2)2,t≥0 \frac{dV}{dt} = \frac{1600}{(4t + 2)^2}, \quad t \ge 0 dtdV=(4t+2)21600,t≥0Given that V=0V = 0V=0 when t=0t = 0t=0:
(i) Solve this differential equation to show that
V=400t2t+1 V = \frac{400t}{2t + 1} V=2t+1400t(ii) Hence find the upper limit to the volume of the balloon.
Find the radius of the balloon at t=4.5t = 4.5t=4.5, giving your answer in m to 3 significant figures.
Find the rate of increase of the radius of the balloon at t=4.5t = 4.5t=4.5, giving your answer to 2 significant figures. Show your working and state the units of your answer.
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.
