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.
