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,

Question

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.

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Bơ 4 years 2021-08-04T23:18:14+00:00 1 Answers 13 views 0

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  1. dangkhoa
    0
    2021-08-04T23:19:30+00:00
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    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

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