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.