Answer to Question #273960 in Calculus for Derek

Question #273960

Pebbles are poured out of a tube at one cubic meter per second. It forms a pile which has the shape of a cone. The height of the cone is equal to the radius of circular base. How fast is the pile of pebbles rising when it is 2 meters high


1
Expert's answer
2021-12-01T17:51:59-0500

Let V = volume of pebble in pile at a time t


"Given\\ \\frac{dv}{dt}=1m^3\/s"


"Find \\ \\frac{dh}{dt}, when \\ h=2m"


"V=\\frac{1}{3}\\pi r^2h"


Diameter = height

2r = h

"r=\\frac{h}{2}"


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


"V=\\frac{1}{12}\\pi h^3"


"\\frac{dv}{dt}=\\frac{1}{12\\pi}(3h^2.\\frac{dh}{dt})"


"\\frac{dv}{dt}=\\frac{3}{12}\\pi h^2 \\frac{dh}{dt}"


because "\\frac{dv}{dt}=1"


"\\frac{3}{12}\\pi h^2 \\frac{dh}{dt}=1"


"\\frac{dh}{dt}=\\frac{12}{3\\pi h^2 }"


"\\frac{dh}{dt}|_{h=2}=\\frac{12}{3\\pi (2)^2 }"


"=0.3183m\/s"

The pile pebbles rise at "0.3183m\/s"




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