The radius of a sphere is increasing at a rate of 5 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 40 mm

Question

The radius of a sphere is increasing at a rate of 5 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 40 mm

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Helga 6 months 2021-07-17T06:42:34+00:00 1 Answers 20 views 0

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    2021-07-17T06:43:53+00:00

    9514 1404 393

    Answer:

      8000π mm^3/s ≈ 25,133 mm^3/s

    Step-by-step explanation:

    The rate of change of volume is found by differentiating the volume formula with respect to time.

      V = 4/3πr^3

      V’ = 4πr^2·r’

    For the given numbers, this is …

      V’ = 4π(20 mm)^2·(5 mm/s) = 8000π mm^3/s ≈ 25,133 mm^3/s

    _____

    Additional comment

    By comparing the derivative to the area formula for a sphere, you see that the rate of change of volume is the product of the area and the rate of change of radius. This sort of relationship will be seen for a number of different shapes.

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