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3 Unit A2 1: Pure Mathematics (A-level only)

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Question 203

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)2

where kkk is a constant.

Given that the temperature of the furnace:

  • is 520∘C520^\circ\text{C}520∘C at the instant the furnace is turned off
  • is 120∘C120^\circ\text{C}120∘C exactly 444 hours after the furnace is turned off
a.

Solve the differential equation to show that, according to the model

θ=at+bct+d \theta = \frac{at + b}{ct + d} θ=ct+dat+b​

where a,b,ca, b, ca,b,c and ddd are integers to be found.

[8]
b.

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.

[3]
Markscheme

3 Unit A2 1: Pure Mathematics (A-level only) Questions

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