To calculate the density of the planet in kilograms per cubic meter ([tex]\(kg/m^3\)[/tex]), we will follow these steps:
1. Convert the radius from kilometers to meters.
2. Convert the mass from grams to kilograms.
3. Compute the volume of the sphere using the formula for the volume of a sphere.
4. Use the mass and volume to find the density.
5. Express the density in standard form with three significant figures.
### Step 1: Convert the radius from kilometers to meters
The radius of the planet is [tex]\( 6371 \)[/tex] km. To convert this to meters, we multiply by [tex]\( 1000 \)[/tex]:
[tex]\[
6371 \text{ km} = 6371 \times 1000 \, \text{m} = 6371000 \, \text{m}
\][/tex]
### Step 2: Convert the mass from grams to kilograms
The mass of the planet is [tex]\( 5.97 \times 10^{27} \)[/tex] grams. To convert this to kilograms, we multiply by [tex]\( 0.001 \)[/tex] (since [tex]\( 1 \)[/tex] gram [tex]\( = 0.001 \)[/tex] kilograms):
[tex]\[
5.97 \times 10^{27} \, \text{g} = 5.97 \times 10^{27} \times 0.001 \, \text{kg} = 5.97 \times 10^{24} \, \text{kg}
\][/tex]
### Step 3: Calculate the volume of the sphere
The volume [tex]\( V \)[/tex] of a sphere is given by the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
Substitute the radius in meters ([tex]\( 6371000 \)[/tex] m):
[tex]\[
V = \frac{4}{3} \pi (6371000)^3 \, \text{m}^3
\][/tex]
Using the provided result, the volume [tex]\( V \)[/tex] is:
[tex]\[
V = 1.0832069168457536 \times 10^{21} \, \text{m}^3
\][/tex]
### Step 4: Calculate the density
Density [tex]\( \rho \)[/tex] is calculated as:
[tex]\[
\rho = \frac{\text{mass}}{\text{volume}}
\][/tex]
Using the mass [tex]\( 5.97 \times 10^{24} \, \text{kg} \)[/tex] and the volume [tex]\( 1.0832069168457536 \times 10^{21} \, \text{m}^3 \)[/tex]:
[tex]\[
\rho = \frac{5.97 \times 10^{24}}{1.0832069168457536 \times 10^{21}} \, \text{kg/m}^3
\][/tex]
Using the provided result, the density [tex]\( \rho \)[/tex] is:
[tex]\[
\rho = 5511.412369286149 \, \text{kg/m}^3
\][/tex]
### Step 5: Express the density in standard form
To express the density in standard form with three significant figures:
[tex]\[
\rho = 5.511 \times 10^3 \, \text{kg/m}^3
\][/tex]
So, the density of the planet is:
[tex]\[
\boxed{5.511 \times 10^3 \, \text{kg/m}^3}
\][/tex]
