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 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.
