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AP Calculus AB· difficulty 3/5

A spherical balloon's volume increases at 20 cm3/s20~\text{cm}^3/\text{s}. When r=5 cmr = 5~\text{cm}, the radius grows at

  • A

    15π cm/s\tfrac{1}{5\pi}~\text{cm/s}

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  • B

    1π cm/s\tfrac{1}{\pi}~\text{cm/s}

  • C

    45π cm/s\tfrac{4}{5\pi}~\text{cm/s}

  • D

    120π cm/s\tfrac{1}{20\pi}~\text{cm/s}

Explanation

V=43πr3V = \tfrac{4}{3}\pi r^3, dV/dt=4πr2dr/dtdV/dt = 4\pi r^2 dr/dt. 20=4π(25)dr/dt=100π dr/dt20 = 4\pi(25) dr/dt = 100\pi\,dr/dt. dr/dt=1/(5π)dr/dt = 1/(5\pi).

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