Ask Your Teacher If a snowball melts so that its surface area decreases at a rate of 10 cm2/min, find the rate at which the diameter decreas

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Ask Your Teacher If a snowball melts so that its surface area decreases at a rate of 10 cm2/min, find the rate at which the diameter decreases when the diameter is 9 cm.

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Aubrey 1 day 2021-10-13T06:02:16+00:00 1 Answer 0

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    2021-10-13T06:03:56+00:00

    Answer:

    The diameter decreases at the rate of 0.1768 cm/min when the diameter is 9 cm.

    Step-by-step explanation:

    A snowball is spherical, so it’s area is given by the following formula:

    A = 4 \pi r^{2}

    The radius is half the diameter so:

    A = 4 \pi (\frac{d}{2})^{2}

    A = 4\pi (\frac{d^{2}}{4})

    A = \pi d^{2}

    If a snowball melts so that its surface area decreases at a rate of 10 cm2/min, find the rate at which the diameter decreases when the diameter is 9 cm.

    This is \frac{dd}{dt} when \frac{dA}{dt} = -10, d = 9

    A = \pi d^{2}

    Applying implicit differentiation:

    We have to variables(A and d), so:

    \frac{dA}{dt} = 2\pi d \frac{dd}{dt}

    -10 = 18 \pi \frac{dd}{dt}

    \frac{dd}{dt} = -\frac{10}{18\pi}

    \frac{dd}{dt} = -0.1768

    The diameter decreases at the rate of 0.1768 cm/min when the diameter is 9 cm.

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