The volume Vcm3 of fluid in a container at time t seconds can be modelled using the differential system dVdt=2V−15. If V=15cm3 at time t=0, calculate the volume of fluid in the container after 2 seconds. Give your answer to the nearest cubic centimetre. Do not include any other symbols or units in your answer.
Question
The ODE is separable, as
dV/dt = 2V – 15 → dV/(2V – 15) = dt
Integrate both sides to get
1/2 ln|2V – 15| = t + C
(To compute integral, consider substituting U = 2V – 15.)
The volume is V = 15 cm³ when t = 0, so that
1/2 ln(15) = C
and so the volume at any time t is such that
1/2 ln|2V – 15| = t + 1/2 ln(15)
Solve for V :
ln|2V – 15| = 2t + ln(15)
2V – 15 = exp(2t + ln(15))
2V – 15 = exp(2t) exp(ln(15))
2V – 15 = 15 exp(2t)
2V = 15 + 15 exp(2t)
V = 15/2 (1 + exp(2t))
(where exp(x) = eˣ )
Then the volume of fluid after t = 2 s is
V = 15/2 (1 + exp(4)) ≈ 417 cm³