a right cone of radius r centimeters and height h centimeters is lowered point f
ID: 2860139 • Letter: A
Question
a right cone of radius r centimeters and height h centimeters is lowered point first at a rate of v cm/s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submergerd?
2. (8 points) A right cone of radius r centimeters and height h centimeters (its volume V = r2/3) is lowered point first at a rate of u cm/s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged?Explanation / Answer
The original Volume of water in the cylinder = R²H......where H is the height of the water level before the cone is immersed.
The Volume of the Cone = r²h.
As the cone is immersed, we can regard the displacement (the change in volume) as being equal to a cone of height h and radius r.
When the cone is fully immersed then h = h and r= r.
The Water Volume thus becomes R²H + r²h....which we can regard as R²H....
where H is the new water level
Then, R²H = R²H + r²h........but h = vt
as we are told that the rate of insertion of the cone = vcm/sec so, R²H = R²H + r²vt ........
which simplifies to, R²H = R²H + r²t
Then, H = H + r²vt/R²
The change in water level = dH/dt = r²v/R²
When the cone just becomes completely submerged then,
r = r and dH/dt = v(r/R)²....which is the speed at which the water level is rising.
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