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    Rate of change and the chain rule

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    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    A rocket ship is speeding away from the sun. In convenient units, the radiation R inside the ship is given by
    R = 1/x^2, where x is the distance to the sun. This distance x, as a function of time t in hours, is given by x = 12t+ 2.

    How far is the ship from the sun when the radiation level is decreasing at a rate of 3 units per hour, when dR/dt =-3?
    What does x=___?

    © BrainMass Inc. brainmass.com December 24, 2021, 10:54 pm ad1c9bdddf
    https://brainmass.com/math/optimization/rate-change-chain-rule-518482

    SOLUTION This solution is FREE courtesy of BrainMass!

    Use the chain rule here:
    dR/dt=dR/dx dx/dt.

    Now, with dR/dx=-2/x^3 and dx/dt=12, then we have the following:
    dR/dt=-24/x^3

    When dR/dt=-3, we see that -3=-24/x^3, so in x^3=8 form, where x=2.

    Alternatively, one can write R in terms of t, by replacing the expression for x:
    R=1/(12t+2)^2
    Then dR/dt=-24/(12t+2)^3, so dR/dt=-3 implies that -24/(12t+2)^3=-3, or equivalently (12t+2)^3=8.
    Solving for t gives only one value, t=0. Replacing the value of t in the expression for x gives x=2.

    Note that the method that uses the chain rule is faster. You may consider that when solving problems of the same kind.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:54 pm ad1c9bdddf>
    https://brainmass.com/math/optimization/rate-change-chain-rule-518482

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