SOLUTION: A container in the shape of a sphere of radius 6cm is filled with water to depth of 10cm. (a)Find the volume of the water (b)If the water flows out through a small hole in the bott

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Question 1189056: A container in the shape of a sphere of radius 6cm is filled with water to depth of 10cm. (a)Find the volume of the water (b)If the water flows out through a small hole in the bottom so that the level drops 2cm , how much water escaped.
Answer by ikleyn(52864) About Me  (Show Source):
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A container in the shape of a sphere of radius 6cm is filled with water to depth of 10cm.
(a) Find the volume of the water
(b) If the water flows out through a small hole in the bottom so that the level drops 2cm, how much water escaped.
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Use the formula for the volume of a spherical cap

    V = %281%2F3%29%2Api%2Ah%5E2%2A%283R-h%29,


where R is the radius of the sphere and h is the height (= the depth) of the cap.


This formula works uninterruptedly in the entire diapason from h= 0 (empty container) to h= 2R (full container).


So, in case (a) you apply the formula at R= 6 cm and h= 10 cm

    V = %281%2F3%29%2A3.14159%2A10%5E2%2A%283%2A6-10%29 = 837.76 cm^3.      ANSWER


I case (b), the final volume is

    V = %281%2F3%29%2A3.14159%2A8%5E2%2A%283%2A6-8%29 = 670.21 cm^3.


The amount of water escaped is the difference  837.76 cm^3 - 670.21 cm^3 = 167.55 cm^3.    ANSWER

Solved.

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For the formulas on the volume of a spherical cap see these Internet sources

https://mathworld.wolfram.com/SphericalCap.html

https://en.wikipedia.org/wiki/Spherical_cap