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 model applies for t>0t>0t>0, with r→0 r\to0\,r→0 as t→0+t\to0^+t→0+. After 5 seconds the radius is 4 cm. Solve the differential equation to find an equation linking r r\,r and ttt.
Suggest one limitation of this model.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.