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31 December, 06:57

If Q represents the quantity of ice crystals measured over time, then calculate Q at t=1 seconds for k=-0.8 / second. Enter your answer in decimal notation (only). Round in the tenths place to give a fractional representation of a partly formed crystal (if any)

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  1. 31 December, 08:05
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    Q (1) = 103.25

    Step-by-step explanation:

    Given:

    - The rate of ice crystals formation/depletion is given by:

    Q' (t) = k * (Q - 70)

    - For the initial conditions;

    Q (0) = 144, k = - 0.8 / sec

    Find:

    Calculate Q at t=1 seconds for k=-0.8 / second. Enter your answer in decimal notation (only).

    Solution:

    - First we will solve the differential equation to get a functions Q with time, Q (t).

    - Use the given ODE:

    dQ / dt = - 0.8 * (Q - 70)

    - Separate variables:

    dQ / (Q - 70) = - 0.8*dt

    - Integrate both sides:

    Ln | Q - 70 | = - 0.8*t + C

    - Use initial conditions to evaluate @ Q (0) = 144:

    Ln | 144 - 70 | = - 0.8*0 + C

    C = Ln | 74 |

    - The Solution is given by:

    Ln | (Q - 70) / 74 | = - 0.8*t

    Q (t) - 70 = 74*e^ (-0.8*t)

    Q (t) = 70 + 74*e^ (-0.8*t)

    - Now use the solution to the ODE and evaluate @ t = 1s

    Q (1) = 70 + 74*e^ (-0.8*1)

    Q (1) = 103.25
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