SOLUTION: A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. Water is flowing into the tank at a rate of 5 cubic feet per minute. Find the rate of change of the de

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Question 1186841: A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. Water is flowing into the tank at a rate of 5 cubic feet per minute. Find the rate of change of the depth of the water when the water is 6 feet deep.
Answer by ikleyn(52792) About Me  (Show Source):
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A conical tank (with vertex down) is 10 feet across the top and 12 feet deep.
Water is flowing into the tank at a rate of 5 cubic feet per minute.
Find the rate of change of the depth of the water when the water is 6 feet deep.
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                 S t e p   b y   s t e p



(1)  The formula for the tank radius as the function of the depth is

         R = %285%2F12%29%2AH.     (1)


     Indeed, it gives the radius  R = 5 ft, when H = 12 ft.



(2)   The formula for the volume of the tank

          V = %281%2F3%29%2Api%2AR%5E2%2AH = %28pi%2F3%29%2A%285%2F12%29%5E2%2AH%28t%29%5E3    (2)

       after substituting (1).



(3)  Differentiate it 

          %28dV%29%2F%28dt%29 = pi%2A%285%2F12%29%5E2%2AH%28t%29%5E2%2A%28%28dH%29%2F%28dt%29%29.



(4)  Substitute H(t) = 6 feet and pi = 3.14159265

           %28dV%29%2F%28dt%29 = 3.14159265%2A%285%2F12%29%5E2%2A6%5E2%2A%28%28dH%29%2F%28dt%29%29 = 19.63495406%2A%28%28dH%29%2F%28dt%29%29.



(5)  Substitute  %28dV%29%2F%28dt%29 = 5 cubic feet per minute

           5 = 19.63495406%2A%28%28dH%29%2F%28dt%29%29.



(6)  From this, find

            %28dH%29%2F%28dt%29 = 5%2F19.63495406 = 0.254647909  feet per minute.



(7)  Round and get the ANSWER:  the rate of change of the depth of the water 
     is  0.25465 feet per minute when the water is 6 feet deep.

Solved.