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A sphere with radius 0.210 m has density rho that decreases with distance r from the center of the sphere according to

Ρ = 2.50 x 10³ kg/m³ - (8.50 x 10³ kg/m⁴)r.
a. Calculate the total mass of the sphere

User Aclowkay
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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

User Anilkumar Patel
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