Math question, plz help?

Question

Grains of fine California beach sand are approximately spheres with an average radius of 36 μ m and are made of silicon dioxide. A solid cube of silicon dioxide with a volume of 1.00 m3 has a mass of 2600 kg. What mass of small sand grains would have a total surface area (the total area of all individual spheres) equal to the surface area of a cube 1.00 m on an edge?

Answer 1

The answer is 187.2 g.

1.  Calculate the surface area of the cube:  The cube has 6 sides, and each side is 1000mm (1 m), so the total surface area of the cube is:

6 x 1000 x 1000 or 6,000,000 mm^2.

2.  Calculate the surface area of an average grain:  The surface area of a single average grain can be represented by the formula for the surface area of a sphere:

SA = 4 x PI x r^2

In this case r is .036 mm (36 μ m) so we have

SA = .016286 mm^2

3.  How many grains are needed to match the surface area?  This means we will need 368,414,220 grains of sand to do the job (6,000,000 / .016286).  But the question didn't ask how many grains were needed, it asked how much mass (weight).  Before we get to weight we must calculate the volume.

4.  How much volume is needed?  The volume of a sphere is represented by the formula

V = 4/3 x PI x r^3

So a single grain has the volume of

4/3 PI (.036mm)^3 or

.000195432 mm^3  This times the number of grains needed (368,414,220) means that we need 72,000 mm^3 of sand.

5.  How much mass is that?  The material weighs 2600kg for every m^3 which equals 2.6 mg/mm^3.  

So the mass needed is the volume needed times the density or

72,000 mm^3 x 2.6 mg/m^3 or

187,000 mg (187.2 g)