A geologist is studying the cooling properties of a basalt sample in a controlled laboratory environment. The sample is heated to a uniform peak temperature of 520∘C520^{\circ}\text{C}520∘C and then placed in a room held at a constant ambient temperature of 20∘C20^{\circ}\text{C}20∘C. At time t t\,t minutes after being placed in the room, the temperature of the basalt, θ∘C\theta^{\circ}\text{C}θ∘C, is observed to decrease at a rate proportional to the difference between its current temperature and the ambient temperature.
Initially, the temperature of the sample is decreasing at a rate of 12.5∘C12.5^{\circ}\text{C}12.5∘C per minute.
Show that
dθdt=−0.025(θ−20) \frac{d\theta}{dt} = -0.025(\theta - 20) dtdθ=−0.025(θ−20)Solve the differential equation
dθdt=−0.025(θ−20) \frac{d\theta}{dt} = -0.025(\theta - 20) dtdθ=−0.025(θ−20)to find an expression for θ \theta\,θ in terms of ttt.
Determine the time taken for the basalt sample to cool to 200∘C200^{\circ}\text{C}200∘C. Give your answer to the nearest minute.
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.
