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AP Calculus AB· difficulty 3/5

Water fills a conical tank (point down) at 4 m3/min4~\text{m}^3/\text{min}. At the moment the water depth is 33 m and surface radius 22 m, the depth rises at

  • A

    1π m/min\tfrac{1}{\pi}~\text{m/min}

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  • B

    44π m/min\tfrac{4}{4\pi}~\text{m/min}

  • C

    49π m/min\tfrac{4}{9\pi}~\text{m/min}

  • D

    14π m/min\tfrac{1}{4\pi}~\text{m/min}

Explanation

By similar triangles r/h=2/3r/h = 2/3, so r=2h/3r = 2h/3. Volume V=13πr2h=4πh327V = \tfrac{1}{3}\pi r^2 h = \tfrac{4\pi h^3}{27}. Then dV/dt=4πh29dh/dtdV/dt = \tfrac{4\pi h^2}{9} dh/dt. At h=3h=3: 4=4π(9)9 dh/dt=4π dh/dt4 = \tfrac{4\pi(9)}{9}\,dh/dt = 4\pi\,dh/dt. So dh/dt=1/πdh/dt = 1/\pi.

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