pre-algebra

Question

I need help with a problem on my worksheet. The problem is,


A caterer wants to have cylinder water glasses completely full, including three spherical ice cubes. The water is going to be poured first and the ice cubes added once the glasses are on the table. The diameter of the water glass is 3 in and the height is 6 in. The diameter of each ice cube is 2 in. How high does the water level need to be so that adding the ice cubes completely fills each glass?

Any way anyone could help? Thank you!

Answer 1

I believe that in this problem the ice 'cubes' are not cubes, but spheres, SINCE THEY HAVE A DIAMETER OF 2".

 

Therefore the volume of the cylindrical glass, (27/2)(pi), is reduced by (4/3)(pi)(1³) for each spherical piece of ice;

three such pieces of ice would account for 4pi cu in to be subtracted from the cylindrical volume.

 

Thus, the volume of water will be 13.5 pi - 4 pi = 9.5 pi.

 

For this volume of water to be put into the cylinder first, we calculate the height as

V = 9.5 pi = pi (3/2)²h; h = 9.5/(1.5²) = 4.2222... inches.

Answer 2

Assuming that the given diameter of glass is inner diameter, the inner volume of glass

=area of base* height

=π* (radius of base)²* height

=π* (half of diameter of base)2* height

=π (3/2)²6 cubic inch

 

=[(π*3*3*6³)/(2*2)]

 

=27π/2 cubic inch

 Diameter of one ice cube being 2 in, volume of one ice cube = (4/3)π(2/2)³ cubic inch = 4π/3 cubic inch

Hence, volume of 3 ice cubes = 4π cubic inch

 

Since each glass will be completely full after adding 3 ice cubes in the glass, the volume of water in each glass before adding 3 ice cubes must be= [(27π/2)-4π] cubic inch =[(27π-8π)/2] cubic inch =19π/2 cubic inch

 

The height of water column in the glass must be = volume of water before adding ice cubes/ area of base of glass

=[19π/2]/[π (3/2)²] inch

 

=[19π*2*2/2*π*3*3] inch

 

=[38/9] inch

 

≈4.222inch

 

Therefore, the water level need to be approximately 4.222 inch high so that adding the ice cubes completely fills each glass.