Illustrate a schematic cross section of the Earth, showing its core and mantle. The core should be depicted as a smaller sphere inside the larger sphere representing the Earth. Provide a graphical representation of the densities and radiuses of the core, mantle and the entire Earth. Make it clear that these densities and radiuses are variable. Please do not include any text.

Using the densities below, find what percentage of the Earth's mass is contained in its core. If Earth's core were adjusted in radius so that the core contained forty percent of the mass, by what factor would the radius of the Earth change compared with its actual radius? (Assume the core and mantle densities remain unchanged as the core volume is changed.)

density of:
core=9710 kg/m^3
mantle=4700 kg/m^3
Earth's mean density=5520 kg/m^3

radius of:
Earth=6400km
core=3500km

Ok so I figured out that 29% of earth's mass is contained in its core.

I just don't know how to proceed with the 2nd part of this question.

Well, if we want to find the percentage of the Earth's mass that is contained in its core, we need to compare the masses of the core and the entire Earth.

The mass of the core can be calculated using the formula:

Mass of core = Volume of core * Density of core

The volume of the core can be calculated using the formula for the volume of a sphere:

Volume of core = (4/3) * pi * (Radius of core)^3

Plugging in the values:

Volume of core = (4/3) * 3.14 * (3500)^3 = 179594850000000 km^3

Now we can calculate the mass of the core:

Mass of core = Volume of core * Density of core = 179594850000000 km^3 * 9710 kg/m^3 = 1743832098500000000 kg

The mass of the Earth can be calculated using the mean density and the formula:

Mass of Earth = Volume of Earth * Density of Earth

The volume of the Earth can be calculated using the formula for the volume of a sphere:

Volume of Earth = (4/3) * pi * (Radius of Earth)^3

Plugging in the values:

Volume of Earth = (4/3) * 3.14 * (6400)^3 = 1.085292288e+12 km^3

Now we can calculate the mass of the Earth:

Mass of Earth = Volume of Earth * Density of Earth = 1.085292288e+12 km^3 * 5520 kg/m^3 = 5.989712588e+24 kg

To find the percentage of the Earth's mass that is contained in its core, we can use the formula:

Percentage = (Mass of core / Mass of Earth) * 100

Percentage = (1743832098500000000 kg / 5.989712588e+24 kg) * 100 = 2.91%

Now, if we adjust the core radius so that it contains 40% of the Earth's mass, we need to find the new radius.

Let's call the new radius "R".

The mass of the core with 40% of the Earth's mass can be calculated using the formula:

Mass of new core = 0.40 * Mass of Earth

The volume of the new core can be calculated using the formula for the volume of a sphere:

Volume of new core = (4/3) * pi * R^3

Now we can calculate the new radius:

R = (Volume of new core / ((4/3) * pi))^(1/3)

Plugging in the values:

Mass of new core = 0.40 * 5.989712588e+24 kg = 2.395885035e+24 kg

Volume of new core = (2.395885035e+24 kg / (9710 kg/m^3)) * (1 / 5520 kg/m^3) = 7.080699355e+19 km^3

R = (7.080699355e+19 km^3 / ((4/3) * 3.14))^(1/3) = 2944 km

Finally, to find the factor by which the radius of the Earth would change, we can divide the new radius by the actual radius:

Factor = R / Actual Radius = 2944 km / 6400 km = 0.46

So, the radius of the Earth would decrease by a factor of 0.46 compared to its actual radius if the core were adjusted to contain 40% of the mass.

To find the percentage of the Earth's mass contained in its core, we need to compare the mass of the core to the total mass of the Earth.

Let's start by finding the volume of the core. The formula for the volume of a sphere is given by:

V = (4/3)πr^3

where V is the volume and r is the radius.

For the core:
V_core = (4/3)π(3500 km)^3

Next, we can calculate the mass of the core by multiplying its volume by its density:

M_core = V_core * density_core

M_core = (4/3)π(3500 km)^3 * 9710 kg/m^3

Now, let's find the total mass of the Earth. We can calculate the volume of the Earth using the same formula as above, but with the Earth's radius:

V_earth = (4/3)π(6400 km)^3

And then multiply the volume by the mean density of the Earth:

M_earth = V_earth * density_earth

M_earth = (4/3)π(6400 km)^3 * 5520 kg/m^3

To find the percentage of the Earth's mass contained in its core, we can divide the mass of the core by the total mass of the Earth and multiply by 100:

Percentage = (M_core / M_earth) * 100

Now let's calculate:

M_core = (4/3)π(3500 km)^3 * 9710 kg/m^3

M_earth = (4/3)π(6400 km)^3 * 5520 kg/m^3

Percentage = (M_core / M_earth) * 100

By adjusting the radius of the Earth's core so that it contains 40% of the mass, we need to find the new radius of the core.

Let's call the new radius r_new.

We can set up an equation using the given 40%:

(M_core_new / M_earth_new) = 0.40

Solving for M_core_new:

M_core_new = 0.40 * M_earth_new

We know that the density of the core remains the same, so the volume of the core remains the same as well:

V_core_new = (4/3)π(r_new)^3

M_core_new = V_core_new * density_core

Substituting the volume expression into the equation, we get:

M_core_new = (4/3)π(r_new)^3 * 9710 kg/m^3

Setting the two equations for M_core_new equal to each other:

V_core_new * density_core = (4/3)π(r_new)^3 * 9710 kg/m^3

Canceling out the constants:

V_core_new = (r_new)^3

Simplifying the equation:

(r_new)^3 * density_core = (r_new)^3 * 9710 kg/m^3

Since the densities are the same, we know that:

(r_new)^3 = (3500 km)^3

Now we can calculate the new radius by taking the cubic root of both sides:

r_new = (3500 km)^(1/3)

To find the factor by which the radius of the Earth would change compared to its actual radius, we can divide the new radius by the actual radius:

Factor = r_new / actual radius

Now, let's calculate the values:

M_core = (4/3)π(3500 km)^3 * 9710 kg/m^3

M_earth = (4/3)π(6400 km)^3 * 5520 kg/m^3

Percentage = (M_core / M_earth) * 100

r_new = (3500 km)^(1/3)

Factor = r_new / actual radius

To find the percentage of Earth's mass contained in its core, we need to calculate the masses of the core and the mantle first.

1. Calculate the volume of the core:
Volume = (4/3) * pi * r^3
Given radius of the core, r = 3500 km = 3500,000 m
Volume = (4/3) * pi * (3500,000)^3 m^3

2. Calculate the mass of the core:
Mass = Volume * Density
Given density of the core = 9710 kg/m^3
Mass of the core = Volume * 9710 kg

3. Calculate the mass of the mantle:
Mass of the mantle = Total mass of the Earth - Mass of the core

To find the total mass of the Earth, we need to calculate its volume first.

Volume of the Earth = (4/3) * pi * r^3
Given radius of the Earth, r = 6400 km = 6400,000 m
Volume of the Earth = (4/3) * pi * (6400,000)^3 m^3

Total mass of the Earth = Volume of the Earth * Mean density of the Earth
= Volume of the Earth * 5520 kg/m^3

Mass of the mantle = Total mass of the Earth - Mass of the core

4. Calculate the percentage of Earth's mass contained in its core:
Percentage = (Mass of the core / Total mass of the Earth) * 100

Now, let's calculate the above values to find the answer.

1. Calculate the volume of the core:
Volume = (4/3) * pi * (3500,000)^3 m^3

2. Calculate the mass of the core:
Mass of the core = Volume * 9710 kg

3. Calculate the mass of the mantle:
Total mass of the Earth = Volume of the Earth * 5520 kg/m^3
Mass of the mantle = Total mass of the Earth - Mass of the core

4. Calculate the percentage of Earth's mass contained in its core:
Percentage = (Mass of the core / Total mass of the Earth) * 100

To find the factor by which the radius of the Earth would change if the core contained forty percent of the mass, we can set up a ratio using the values from the calculations above:

Factor = (New radius of the Earth) / (Actual radius of the Earth)

We know that the core mass should be 40% of the total mass, so:

(Mass of the core / Total mass of the Earth) = 0.40

With this information, we can find the new radius of the Earth:

1. Calculate the mass of the core and the total mass of the Earth as done above.

2. Solve the equation (Mass of the core / Total mass of the Earth) = 0.40 for the new radius of the Earth.

3. Plug the new radius into the ratio formula:

Factor = (New radius of the Earth) / (Actual radius of the Earth)

Calculating the above steps will give you the percentage of Earth's mass contained in its core and the factor by which the radius of the Earth would change if the core contained forty percent of the mass.