Final answer:
To calculate the total mass of the sphere, we can integrate the given density function over the sphere's volume. The volume of a sphere with radius 0.210 m can be calculated using the formula V = 4/3πr³. Integrating the density function over this volume will give us the total mass of the sphere, which is approximately 0.271 kg.
Step-by-step explanation:
The total mass of the sphere can be calculated by integrating the density function over the entire volume of the sphere.
We have the density function given as
Ρ = (2.50 x 10³ kg/m³) - (8.50 x 10³ kg/m⁴)r.
To find the total mass of the sphere, we need to determine the volume of the sphere with radius 0.210 m and then integrate the density function over that volume.
The volume of a sphere is given by the formula V = 4/3πr³.
Using this formula, we can calculate the volume of the sphere with radius 0.210 m.
V = 4/3π(0.210 m)³ ≈ 0.03874 m³
Now, we can integrate the density function over the volume of the sphere to find the total mass.
Mass = ∫(2.50 x 10³ kg/m³ - (8.50 x 10³ kg/m⁴)r) dV
Mass = ∫(2.50 x 10³ kg/m³ - (8.50 x 10³ kg/m⁴)r) d(4/3πr³)
Mass = ∫(2.50 x 10³ kg/m³ - (8.50 x 10³ kg/m⁴)r) (4/3πr³) dr
Mass = (2.50 x 10³ kg/m³)(4/3π)∫r³ dr - (8.50 x 10³ kg/m⁴)(4/3π)∫r⁴ dr
Mass = (2.50 x 10³ kg/m³)(4/3π)(1/4)r⁴ - (8.50 x 10³ kg/m⁴)(4/3π)(1/5)r⁵
Mass = (2.50 x 10³ kg/m³)(π/3)r⁴ - (8.50 x 10³ kg/m⁴)(π/15)r⁵
Substituting the limits of integration from 0 to 0.210 m, we can evaluate the integral:
Mass = (2.50 x 10³ kg/m³)(π/3)(0.210 m)⁴ - (8.50 x 10³ kg/m⁴)(π/15)(0.210 m)⁵
Mass ≈ 0.271 kg