Question

A hemispherical bowl is being filled with water at a uniform rate. When the height of the water is h cm, the volume is [(pi)(r(h^2) - (1/3)(h^3))]cm^3, r cm being the radius of the hemisphere. Find the rate at which the water level is rising when it is half way to the top, given that r = 6 and that the bowl fills in 1 min.

Answer 1

the volume of the sphere is \[V _{sphere}=\frac{ 4 }{ 3 }\pi r ^{3}\]
therefore the volume of hemisphere is \[V _{hemisphere} = \frac{ 2 }{ 3 }\pi r ^{3}\]
and since the radius r = 6 cm and time to fill the bowl is t = 1 min... the rate at which the bowl is filled can be computed...
\[V _{hemisphere} = \frac{ 2 }{ 3 }\pi \left( 6 \right)^{3}=144\pi\] in cm^3
\[r = \frac{ V _{hemispher} }{ t }=\frac{ 144\pi }{ 1 }=144\pi\] in cm^3/minute.

Answer 2

the equation for the volume of the bowl related to height h (cm.) of the liquid poured in the bowl is given as....\[V _{bowl}=\pi \left( rh ^{2}-\frac{ 1 }{ 3 }h ^{3} \right)=\pi rh ^{2}-\frac{ \pi }{ 3 }h ^{3}\]

Answer 3

since this is a time rate problem in Calculus, we can get the 1st derivative of the given equation, that is term by term... we differentiate it to time parameter t...
\[\frac{ dV _{bowl} }{ dt }=(2\pi rh-\pi h^{2})\frac{ dh }{ dt }\]

Answer 4

The problem states that "Find the rate at which the water level is rising when it is half way to the top, given that r = 6 ..." therefore the height h = 3cm. The rate of filling the bowl is already computed, that is... \[\frac{ dV _{bowl} }{ dt }=144\pi \frac{ cm ^{3} }{ \min. }\]

Answer 5

so the rate at which the water level rises is...
\[144\pi=[2\pi(6)(3)-\pi (3)^{2}]\frac{ dh }{ dt }\]
\[\frac{ dh }{ dt }=\frac{ 144\pi }{ \pi(32-9) }=\frac{ 144 }{ 23 }=6.261\frac{ cm }{ \min. }\]

Answer 6

@fatma01 as much as possible i don't want to solve it, i want you to solve it by just giving hints and tips to solve the problem... but since you are not around... i solved it providing explanations to solutions... hope you will understand how did i arrived at obtained answer... :-)

Answer 7

why exactly is the height 3cm i don't think i got that part.

Answer 8

from your problem, it says, "... when it is half way to the top,... " since the radius of the bowl is 6cm from the top to bottom, half way means h=3cm... half of 6cm... :)