A high-performance server processor is equipped with a failsafe cooling system. When the system is shut down at time t=0t = 0t=0 minutes, the processor temperature, θ∘C\theta^\circ\text{C}θ∘C, decreases at a rate proportional to the difference between the processor's current temperature and the ambient room temperature, which is a constant 25∘C25^\circ\text{C}25∘C.
Immediately upon shutdown, the processor is at 145∘C145^\circ\text{C}145∘C and its temperature is decreasing at a rate of 3∘C3^\circ\text{C}3∘C per minute.
Show that
dθdt=−0.025(θ−25) \frac{d\theta}{dt} = -0.025(\theta - 25) dtdθ=−0.025(θ−25)Solve the differential equation
dθdt=−0.025(θ−25) \frac{d\theta}{dt} = -0.025(\theta - 25) dtdθ=−0.025(θ−25)to find an expression for θ\thetaθ in terms of ttt.
Determine the time taken for the processor temperature to drop to 85∘C85^\circ\text{C}85∘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.
