What (decimal) fraction of a hypothetical spherical planet's volume does a planetary core with a radius of 70% of that planet's radius have? Round off answer to three decimal places. [Hint: the volume of a sphere is equal to 4/3 pi times the radius cubed]
(0.7)^3 = 0.343
To find the fraction of a hypothetical spherical planet's volume that a planetary core with a radius of 70% of that planet's radius has, we need to compare the volume of the core to the volume of the entire planet.
Let's assume the radius of the spherical planet is represented by "r".
The volume of a sphere is given by the formula: V = (4/3) * π * r^3.
Therefore, the volume of the planet is V = (4/3) * π * r^3.
Now, let's calculate the volume of the core. Given that the core has a radius of 70% of the planet's radius, we can say that the radius of the core is 0.7 * r.
The volume of the core is V_core = (4/3) * π * (0.7 * r)^3.
To find the fraction of the planet's volume occupied by the core, we divide the volume of the core by the volume of the planet:
Fraction = V_core / V = [(4/3) * π * (0.7 * r)^3] / [(4/3) * π * r^3]
Simplifying the expression:
Fraction = [(0.7 * r)^3] / [r^3]
Fraction = (0.7)^3
Calculating (0.7)^3:
Fraction = 0.7 * 0.7 * 0.7 = 0.343
Rounding off the answer to three decimal places, the fraction of the planet's volume occupied by the core is approximately 0.343.