SOLUTION: The water in a hemi-spherical bowl is 42 cm across the top is 9 cm deep. How much more water is needed to fill the bowl to the brim?

Here is everthing you need:

The water in the bottom of the hemispheric bowl is a spherical cap, and the
formula for its volume is

Volume of a spherical cap = , where h=9 cm. is the
height of the spherical cap (water), and radius R = 21 cm. (half the 42 cm
diameter at the top).

Volume of a sphere is , where R = 21 cm. 

This is a hemisphere, not a whole sphere, so remember that the volume of a
hemisphere is only 1/2 the volume of a whole sphere.

So you only need to calculate those two volumes and subtract them to find the
volume to be filled between the top of the water and the top of the bowl.

Your answer will be in cubic centimeters. Then you can use the fact that 
1 cubic centimeter of water is exactly 1 milliliter of water.  Then to 
convert from milliliters to liters, you will multiply by 1000.  

Edwin

If you make a sketch, you will see a right angled triangle.

Its hypotenuse is the radius of the sphere R.

Its legs are  (R-9) cm and 42/2 = 21 cm.

So, we write the Pythagorean equation

    (R-9)^2 + 21^2 = R^2.


From it, we find

    R^2 - 18R + 81 + 441 = R^2,

    81 + 441 = 18R

    18R = 522

      R = 522/18 = 29.


Thus the sphere radius is 29 cm.


Then we find the volume of the hemi-sphere

     =  =  = 51080.202 cm^3.


Next you find the volume of the spherical segment for R = 29 cm and h = 9 cm
using the formula

     =  =  = 6616.194 cm^3.


Now your answer is the difference of the two volumes

     -  = 51080.202 - 6616.194 = 44464.008 cm^3,  or 44.464 liters  (rounded).

Ikleyn and I both interpreted the 9 cm the same way, as the depth of the water
in the bowl.  However, I interpreted the 42 cm as the diameter of the bowl and
Ikleyn interpreted the 42 cm as the diameter of the surface of the water, not
the bowl.  
Let's analyze this sentence as it was posted:

First of all, the sentence is very ungrammatical. It is of the form

"X is Y is Z".

Such is a bad violation of English grammar.

The way I interpreted it was simply by omitting the first is.

Simply omitting that first is makes it grammatically correct and perfectly 
clear and understandable as the way I interpreted it.

Ikleyn considered the sentence as though there were a period after the word "top".  
In other words, she interpreted it as if it had been written like this:

Interpreting the sentence that way leaves the last part 

"is 9 cm deep" 

dangling in the air, as just a predicate with no grammatical subject.
 
I maintain that mine is the more obvious way to interpret the ungrammatical
thing that was posted. 

We keep getting posts with bad English grammar.  I suspect it is because these
are being translated from another language into English by someone whose first-
learned language was NOT English. 

Edwin