Find the mass of the solid whose density is equal to twice the distance from the origin, which is outside the sphere of radius 3 and inside the sphere of radius 5.
A) $544 \pi$
B) $1088 \pi$
C) $272 \pi$
D) $136 \pi$

Question

Quizplus · Accepted Answer

The mass of the solid can be calculated using the formula for the volume of a spherical shell and the given density function. The density is given as $2r$, where $r$ is the distance from the origin. The volume of a spherical shell is given by the difference in volumes of the two spheres, $V = \frac{4}{3}\pi(5^3 - 3^3)$. To find the mass, integrate the density function over the volume of the shell. The mass $M$ is given by the integral of the density times the differential volume element in spherical coordinates, leading to $M = \int_{3}^{5} 2r \cdot 4\pi r^2 dr$, which simplifies to $1088 \pi$.

Find the Mass of the Solid Whose Density Is Equal 544π544 \pi544π

Find the Mass of the Solid Whose Density Is Equal $544 \pi$