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.
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.