Solve the problem.
-Find the mass of the solid in the first octant between the spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 100$ if the density at any point is inversely proportional to its distance from the origin.
A) $28 \mathrm { k } \pi$
B) $168 \mathrm { k } \pi$
C) $14 \mathrm { k } \pi$
D) $21 \mathrm { k } \pi$

Question

Quizplus · Accepted Answer

Let the density at point $(x,y,z)$ be $\delta(x,y,z)$. Then we have $$\delta(x,y,z)=\frac{k}{\sqrt{x^2+y^2+z^2}}$$ where $k$ is a constant of proportionality. Then, the mass of an infinitesimal element of the solid is $\delta(x,y,z) dV$ where $dV=dxdydz$. Using spherical coordinates, we have $$dV=r^2\sin	heta drd	heta d\phi$$ where $r$ is the distance from the origin, $	heta$ is the polar angle, and $\phi$ is the azimuthal angle. The volume element is confined to the first octant, so we have $0\leq	heta\leq\frac{\pi}{2}$ and $0\leq\phi\leq\frac{\pi}{2}$. Therefore, we have
\begin{align*}
	ext{Mass}&=\iiint\limits_{	ext{Solid}}\delta(x,y,z)dV\
&=\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\int_{4}^{10}\frac{k}{\sqrt{r^2}}r^2\sin	heta drd	heta d\phi\
&=4k\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{\pi}{2}}\int_{4}^{10}r\sin	heta drd	heta d\phi\
&=4k\left(\int_{0}^{\frac{\pi}{2}}\sin	heta d	hetaight)\left(\int_{0}^{\frac{\pi}{2}}d\phiight)\left(\int_{4}^{10}r dr ight)\
&=4k\cdot 1\cdot\frac{\pi}{2}\cdot\frac{1}{2}(10^2-4^2)\
&=21k\pi
\end{align*}
Thus, the mass of the solid is given by $21k\pi$. To find $k$, we note that the density at a point on the sphere $x^2+y^2+z^2=16$ is $\frac{k}{\sqrt{16}}=\frac{k}{4}$ and the density at a point on the sphere $x^2+y^2+z^2=100$ is $\frac{k}{\sqrt{100}}=\frac{k}{10}$. Since the density is inversely proportional to the distance from the origin, we have $$\frac{\frac{k}{4}}{\frac{k}{10}}=\frac{10}{4}$$ which yields $k=5$. Therefore, the total mass is $21(5)\pi=\boxed{	extbf{(D) }21k\pi}$.

Solve the Problem x2+y2+z2=16x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16x2+y2+z2=16

Solve the Problem $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$