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AP Calculus AB· difficulty 4/5

Water drains from an inverted cone (height 12 m, base radius 6 m) at a constant rate of 2 m3/min2\,\text{m}^3/\text{min}. How fast is the water level falling when the depth is 4 m?

  • A

    18π\frac{1}{8\pi} m/min

  • B

    14π\frac{1}{4\pi} m/min

  • C

    2π\frac{2}{\pi} m/min

  • D

    12π\frac{1}{2\pi} m/min

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Explanation

Similar triangles: r/h=6/12=1/2r/h = 6/12 = 1/2, so r=h/2r=h/2, V=13πr2h=πh312V = \frac{1}{3}\pi r^2 h = \frac{\pi h^3}{12}. dV/dt=πh24⋅dh/dtdV/dt = \frac{\pi h^2}{4} \cdot dh/dt. At h=4h=4: −2=π⋅164dh/dt=4π⋅dh/dt-2 = \frac{\pi \cdot 16}{4} dh/dt = 4\pi \cdot dh/dt → dh/dt=−1/(2π)dh/dt = -1/(2\pi). Magnitude 1/(2π)1/(2\pi) m/min.

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