Math Is Fun Forum

ose90

Member

Registered: 2008-09-20

Posts: 3

Application of diferentiation, help me

A hemisphere bowl of radius 12cm is initially full of water. Water runs out of a small hole at the bottom of the bowl at a rate of 48pi cm^3 s^-1. When the depth of the water is x cm , show that the depth is decreasing at a rate of 48/[x(24-x)] cm s^-1
Also, find the rate at which the depth is decreasing when

a) The bowl is full.
b)The depth is 6cm.

Another question is in this picture

Thanks in advance! Really urgent

Last edited by ose90 (2008-09-20 17:00:49)

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Application of diferentiation, help me

imagine the right side of a semi-circle in 2D displaced vertically so the bottom rests at origin, it would have equation

rotating the region formed by right hand side's integral to a height 'h' around y axis gives the volume of hemisphere to a certain depth

Since we are looking for derivitive we don't need to evaluate this (although i think it's interesting to do anyways :P) i'm simply showing it to verify my next step of writing:

  where x is the depth as in your question (not my first equation), and i've subbed the value for the radius in.

the bowl is full; x = 12cm, dx/dt = 1/3 cms-¹
depth is 6cm, x = 6cm, dx/dt = 4/9 cms-¹

Last edited by luca-deltodesco (2008-09-20 20:24:16)

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ose90

Member

Registered: 2008-09-20

Posts: 3

Re: Application of diferentiation, help me

Thanks for your elaborated explanation. I have understood it in a more detailed way,thanks!

Would you mind helping me with the 2nd question?

REgards

ose90

Member

Registered: 2008-09-20

Posts: 3

Re: Application of diferentiation, help me

Thanks you very much, glad to inform you that I have solved both questions.

Regards,
Frank