Find the mass of a solid hemisphere of radius 5 if the mass density at any point on the solid is directly proportional to its distance from the base of the solid.
A) $\frac { 625 } { 4 }$ k $ \pi$
B) $25$ k $ \pi$
C) $\frac { 25 } { 4 }$ k $ \pi$
D) $\frac { 125 } { 4 }$ k $ \pi$

Question

Quizplus · Accepted Answer

The mass density at any point on the solid is directly proportional to its distance from the base of the solid. So, the mass density at a distance $r$ from the base is $kr$, where $k$ is a constant.The volume of a solid hemisphere of radius $r$ is $\frac{2}{3}\pi r^3$.So, the mass of the solid hemisphere is$$m=\int_0^5 kr\cdot \frac{2}{3}\pi r^3 dr=\frac{2}{3}\pi k\int_0^5 r^4 dr=\frac{2}{3}\pi k\left[\frac{r^5}{5}\right]_0^5=\frac{2}{3}\pi k\left(\frac{5^5}{5}-0\right)=\frac{2}{3}\pi k(625)=\frac{625}{4}k\pi$$

Find the Mass of a Solid Hemisphere of Radius 5 6254\frac { 625 } { 4 }4625​

Find the Mass of a Solid Hemisphere of Radius 5 $\frac { 625 } { 4 }$