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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.