Calculus A conical container with a diameter of 30 cm with a height of 45 cm being filled with water at a rate if 16 cm^3.Find the rate of change at which the water is rising in the cylinder when...

Calculus

  1. A conical container with a diameter of 30 cm with a height of 45 cm being filled with water at a rate if 16 cm^3.Find the rate of change at which the water is rising in the cylinder when volume reaches 8(Pi)cm^3.

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thilina-g | College Teacher | (Level 1) Educator

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The container is an upside down cone.

The height, `H = 45 cm`

The area of the top of cone, `A = pi(15)^2 = 225pi`

If the water height at time t is h and diameter is 2r``.

The water volume filled at time `v = 1/3pir^2h`

But according to geometrical analysis we know,

`r/h = 15/45`

`r/h = 1/3`

`h = 3r`

So we get, `v = 1/3pir^2(3r) = pir^3`

`v = pir^3`

We know `(dv)/(dt) = 16` (The filling rate)

Therefore,

`16 = (dv)/(dt) = 3pir^2(dr)/(dt)`

`(dr)/(dt) = 16/(3pir^2)`

The radius on top of water when `v = 8pi` is,

`8pi = pir^3`

`r = 2` cm.

Therefore when `v = 8pi` , r is equal to 2 cm.

Therefore `(dr)/(dt) = 16/(3pi2^2)`

`(dr)/(dt) = 0.424` cm per second


But we know, `h =3r`

Then,

`(dh)/(dt) = 3(dr)/(dt)`

`(dh)/(dt) = 3(0.424)` cm per second

 

Therefore the rate of height increase or the rate at water level changes  is 1.272 cm per second.

 

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