Water is slowly leaking from a bird bath at the rate of 8 cm3/min. The birdbath is in the shape of a cone with the vertex down and the radiu

Question

Water is slowly leaking from a bird bath at the rate of 8 cm3/min. The birdbath is in the shape of a cone with the vertex down and the radius is double the height. How fast is the water level changing when the water is 10 cm deep

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Edana Edana 3 years 2021-07-20T14:55:26+00:00 1 Answers 13 views 0

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  1. thucuc
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    2021-07-20T14:57:00+00:00
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    Answer:

    The water level is changing at – 0.016 centimeters per minute when the water is 10 centimeters deep.

    Step-by-step explanation:

    From Geometry, the volume of the cone (V), measured in cubic centimeters, is defined by:

    V = \frac{\pi\cdot r^{2}\cdot h}{3} (1)

    Where:

    r – Radius, measured in centimeters.

    h – Height, measured in centimeters.

    Then, we find a formula for the rate of change of the volume of the birth bath (\dot V), measured in cubic centimeters per minute, by means of differentiation:

    \dot V = \frac{\pi}{3}\cdot \left(2\cdot r\cdot h\cdot \dot r + r^{2}\cdot \dot h \right) (2)

    Where:

    \dot r – Rate of change of radius, measured in centimeters per minute.

    \dot h – Rate of change of height, measured in centimeters per minute.

    In addition, we have the following relationship:

    r = 2\cdot h (3)

    And by Differential Calculus:

    \dot r = 2\cdot \dot h (4)

    By applying (3) and (4) in (2), we find the following expanded formula:

    \dot V = \frac{\pi}{3}\cdot \left[2\cdot (2\cdot h)\cdot (2\cdot \dot h)+4\cdot h^{2}\cdot \dot h\right]

    \dot V = \frac{\pi}{3}\cdot (8\cdot h+4\cdot h^{2})\cdot \dot h (5)

    If \dot V = -8\,\frac{cm^{3}}{min} and h = 10\,cm, then the rate of change of the water level is:

    \dot h = \frac{\dot V}{\frac{\pi}{3}\cdot (8\cdot h + 4\cdot h^{2}) }

    \dot h = \frac{-8\,\frac{cm^{3}}{min} }{\frac{\pi}{3}\cdot [8\cdot (10\,cm)+4\cdot (10\,cm)^{2}] }

    \dot h \approx -0.016\,\frac{cm}{min}

    The water level is changing at – 0.016 centimeters per minute when the water is 10 centimeters deep.

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