Answer to Question #330030 in Calculus for Shikongo

Question #330030

A cone of radius r centimetres and height h centimetres iss lowered first at a rate of 1 cm/s into a tall cylinder of radius R that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged?


1
Expert's answer
2022-04-19T02:36:30-0400

"z\\left( t \\right) -coordinate\\,\\,of\\,\\,the\\,\\,cone\\,\\,vertex\\\\z\\left( 0 \\right) =0\\\\z'\\left( t \\right) =-1\\\\y\\left( t \\right) \\,\\,-\\,\\,the\\,\\,water\\,\\,level\\\\y\\left( 0 \\right) =0\\\\\\pi R^2y\\left( t \\right) \\,\\,-\\,\\,volume\\,\\,of\\,\\,the\\,\\,cylinder\\\\\\frac{1}{3}\\pi \\left( y\\left( t \\right) -z\\left( t \\right) \\right) \\left( \\frac{y\\left( t \\right) -z\\left( t \\right)}{h}r \\right) ^2-volume\\,\\,of\\,\\,the\\,\\,part\\,\\,of\\,\\,cone\\,\\,in\\,\\,water\\\\\\pi R^2y\\left( t \\right) -\\frac{\\pi r^2}{3h^2}\\left( y\\left( t \\right) -z\\left( t \\right) \\right) ^3=0\\\\Completely\\,\\,submerged:\\\\y\\left( t \\right) -z\\left( t \\right) =h\\\\\\pi R^2y'-\\frac{\\pi r^2}{3h^2}3\\left( y'-z' \\right) \\left( y\\left( t \\right) -z\\left( t \\right) \\right) ^2=0\\\\\\pi R^2y'-\\pi r^2\\left( y'+1 \\right) =0\\Rightarrow y'=\\frac{r^2}{R^2-r^2}"


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