The volume of a ball is given in terms of its radius by V=4π3R3. Use this to find the average density of Saturn, i.e. the mass per unit volume. Use Earth’s mass in kg from the Table of Constants to express your answer in kgm3. Note for comparison that water has a density of about 1,000 kgm3. Please round your answer to two significant digits.
To find the average density of Saturn, we need to use the given volume equation and the mass of Saturn. The volume equation is provided as V = 4π/3 * R^3.
Step 1: Find the volume of Saturn.
The volume equation of a ball is given as V = 4π/3 * R^3, where R is the radius of the ball. However, the radius of Saturn is not given. We can still find the volume using the average radius, which is about 58,232 kilometers (km) or 5.8232 x 10^7 meters (m).
V = 4π/3 * (5.8232 x 10^7)^3
Step 2: Convert the volume from m^3 to km^3.
Since the volume is given in m^3, we need to convert it to km^3 for ease of comparison with other densities. To do this, divide the volume by (1,000)^3.
V = (4π/3 * (5.8232 x 10^7)^3) / (1,000)^3
Step 3: Find the mass of Saturn.
The mass of Earth is given as 5.972 x 10^24 kg in the Table of Constants. The mass of Saturn is about 95.16 times the mass of Earth. Therefore, the mass of Saturn is:
Mass of Saturn = 95.16 * (5.972 x 10^24) kg
Step 4: Calculate the average density of Saturn.
To find the average density, divide the mass of Saturn by its converted volume.
Density = (Mass of Saturn) / (Volume of Saturn)
Now, let's substitute the values into the equation and calculate:
Density = ((95.16 * (5.972 x 10^24)) / ((4π/3 * (5.8232 x 10^7)^3) / (1,000)^3))
Using a calculator, we can simplify and compute the value. Rounding to two significant digits and expressing the answer in kg/m^3 gives us the average density of Saturn.