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 AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions 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), matched to the AQA A Level Maths (7357) 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.
