As mentioned before, our asteroid is in the shape of a sphere and has a mass of 1000 kilograms. Determine the density (in grams per cubic centimeter) of this asteroid if its diameter is known to be 1.2 meters. Useful information: 1 kg = 1000 g, 1 m = 100 cm, volume of sphere = 4/3 ? r3. Remember that the radius of a sphere is equal to half its diameter. Show all of your work. (20 points)

Density [tex]D[/tex] is a characteristic property of a material and is defined as the relationship between the mass [tex]m[/tex] and volume [tex]V[/tex] of a specific substance or material. So, the density of the asteroid is given by the following equation:

[tex]D=\frac{m}{V}[/tex] (1)

On the other hand, we know the asteroid has a mass [tex]m=1000kg[/tex] and is spherical. This means its volume is given by the following formula:

[tex]V=\frac{4}{3}}\pi r^{3}[/tex] (2)

Where [tex]r=\frac{d}{2}=\frac{1.2m}{2}=0.6m[/tex] is the radius of the sphere and is half its diameter [tex]d[/tex].

Knowing this, we can calculate the volume:

[tex]V=\frac{4}{3}}\pi (0.6m)^{3}[/tex] (3)

[tex]V=0.904m^{3}[/tex] (4)

Substituting (4) in (1):

[tex]D=\frac{1000kg}{0.904m^{3}}=1105.242\frac{kg}{m^{3}}[/tex] (5) This is the density of the asteroid, but we were asked to find it in [tex]\frac{g}{cm^{3}}[/tex]. This means we have to make the conversion:

[tex]D=1105.242\frac{kg}{m^{3}}.\frac{1000g}{1kg}.\frac{1m^{3}}{(100cm)^{3}}[/tex]

Finally:

[tex]D=1.1052\frac{g}{cm^{3}}[/tex]