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1.7.20 Constructing differential equations (A-level only)

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Question 11

The pressure, PPP kPa, inside a leaking decompression chamber ttt minutes after the leak is first detected, is modelled by the differential equation

dPdt=−k(P−100)2 \frac{\text{d}P}{\text{d}t} = -k(P - 100)^2 dtdP​=−k(P−100)2

where kkk is a positive constant and P>100P > 100P>100.

Given that the pressure in the chamber:

  • is 110011001100 kPa at the instant the leak is detected
  • is 350350350 kPa exactly 101010 minutes later
a.

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

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

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

[6]
b.

Hence find, according to the model, the time taken for the pressure in the chamber to reach 150150150 kPa. Give your answer to the nearest minute.

[3]

1.7.20 Constructing differential equations (A-level only) Questions

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