The rate of change of the temperature of a heated metal rod (TTT) in a laboratory setting is directly proportional to the difference between the temperature of the rod and the constant room temperature (15∘C15^{\circ}\text{C}15∘C).
Write down a differential equation for this relationship.
Show that T=15+AektT = 15 + Ae^{kt}T=15+Aekt where A A\,A and k k\,k are constants.
Given that the initial temperature of the rod is 115∘C115^{\circ}\text{C}115∘C, write down the value of AAA.
After 10 minutes, the temperature of the rod is 65∘C65^{\circ}\text{C}65∘C. Show that k=−110ln2\displaystyle k = -\frac{1}{10} \ln 2k=−101ln2.
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.