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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.