The temperature, θ∘C\theta^\circ\text{C}θ∘C, of a industrial furnace, ttt hours after it is switched off, is modelled by the differential equation
dθdt=−k(θ−20)2 \frac{\text{d}\theta}{\text{d}t} = -k(\theta - 20)^2 dtdθ=−k(θ−20)2where kkk is a constant.
Given that the temperature of the furnace:
Solve the differential equation to show that, according to the model
θ=at+bct+d \theta = \frac{at + b}{ct + d} θ=ct+dat+bwhere a,b,ca, b, ca,b,c and ddd are integers to be found.
Hence find, according to the model, the time taken for the temperature of the furnace to reach 45∘C45^\circ\text{C}45∘C. Give your answer to the nearest hour.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions 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, matched to the Edexcel A Level Maths (9MA0) 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.
