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25 June, 02:52

Recall that the volume V of a sphere of radius r can be computed using the formula. If the radius of a spherical balloon is increasing at a rate of 5 cm/min, how fast is its volume increasing when the radius is 10 cm?

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  1. 25 June, 04:15
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    Given that,

    A sphere has a volume of V and a radius r

    Volume of sphere cam be determine using the formula

    V = 4/3 πr³

    V = 4πr³ / 3.

    If the radius of the sphere is increasing by

    dr / dt = 5cm / min

    How fast is the volume increasing when r = 10cm

    dV / dt = ?

    From V = 4πr³ / 3

    We can calculate dV/dr

    dV/dr = 12πr² / 3

    dV/dr = 4πr²

    Then,

    We want to find dV/dt

    Using chain rule

    dV/dt = dV/dr * dr / dt

    dV/dt = 4πr² * 5

    dV/dt = 20πr²

    So, at r = 10

    dV/dt { 20π * 10²

    dV/dt = 6283.19 cm/min

    The rate at which the volume increase is 6283.19 cm/min
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