A spherical balloon is being inflated. At time t t\,t seconds the balloon has radius r r\,r cm and volume V V\,V cm3^33, where V=43πr3\displaystyle V = \frac{4}{3}\pi r^3V=34πr3.
The volume of the balloon is modelled as increasing at a constant rate.
Show that
drdt=kr2\displaystyle \frac{dr}{dt} = \frac{k}{r^2}dtdr=r2k
where k k\,k is a positive constant.
The balloon is initially empty, and after 5 seconds its radius is 4 cm. Solve the differential equation to find an equation linking r r\,r and ttt.
Suggest one limitation of this model.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, 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). Each one has a worked solution and a mark scheme showing where the marks go.