A newly discovered giant planet has an aver-
age radius 12 times that of the Earth and a
mass 722 times that of the Earth.
Calculate the ratio of the new planet’s den-
sity to the Earth’s density.
To find the ratio of the new planet's density to the Earth's density, we need to calculate the densities of both planets.
Density is defined as mass divided by volume:
density = mass/volume
Let's denote the mass of the new planet as M and its average radius as R. Similarly, let's denote the mass of the Earth as Mᴇ and its radius as Rᴇ.
The volume of a sphere is given by the formula:
volume = (4/3)πr³
For the new planet, the volume can be expressed as:
volume = (4/3)πR³
Thus, the density of the new planet, D, can be calculated as:
D = M/volume
= M/[(4/3)πR³]
The Earth's density, Dᴇ, can be calculated in the same way:
Dᴇ = Mᴇ/[(4/3)πRᴇ³]
Now we can find the ratio of the new planet's density to the Earth's density:
D/Dᴇ = (M/[(4/3)πR³]) / (Mᴇ/[(4/3)πRᴇ³])
Canceling out common terms:
D/Dᴇ = (M/Mᴇ) * (Rᴇ³/R³)
Given that the mass of the new planet is 722 times that of the Earth, and the radius is 12 times that of the Earth, we can substitute these values into the equation:
D/Dᴇ = (722) * (1/12³)
Calculating:
D/Dᴇ = (722) * (1/1728)
≈ 0.41875
Therefore, the ratio of the new planet's density to the Earth's density is approximately 0.41875.
To calculate the ratio of the new planet's density to the Earth's density, we need to first find the density of both objects.
Density is defined as mass divided by volume:
Density = Mass / Volume
Let's start with the Earth's density:
1. The mass of the Earth is given as 722 times the mass of the Earth, so we can represent it as Mass(Earth) = 1 Earth Mass.
2. The volume of the Earth can be calculated using the formula for the volume of a sphere: V = (4/3) * π * r^3, where r is the radius.
3. The average radius of the Earth is approximately 6,371 km.
4. Calculate the Earth's volume using the formula: V(Earth) = (4/3) * π * (6371 km)^3
Now, let's calculate the density of the Earth:
Density(Earth) = Mass(Earth) / Volume(Earth)
For the new planet, we follow the same steps:
1. The mass of the new planet is given as 722 times the mass of the Earth, so we can represent it as Mass(New Planet) = 722 Earth Masses.
2. The average radius of the new planet is given as 12 times the radius of the Earth, so we can represent it as Radius(New Planet) = 12 * 6371 km.
3. Calculate the volume of the new planet using the formula: V(New Planet) = (4/3) * π * (12 * 6371 km)^3
Now, let's calculate the density of the new planet:
Density(New Planet) = Mass(New Planet) / Volume(New Planet)
Finally, we can calculate the ratio of the new planet's density to the Earth's density:
Ratio = Density(New Planet) / Density(Earth)