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.
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.