Solve the problem.
-Find the center of mass of the solid enclosed between the cone with equation $z = \frac { 7 } { 8 } \sqrt { x ^ { 2 } + y ^ { 2 } }$ and the plane with equation $z = 7$ if the density at any point is proportional to the distance from that point to the plane $z = 7$.
A) $( \bar { x } , \bar { y } , \bar { z } ) = \left( 0,0 , \frac { 21 } { 5 } \right)$
B) $( \overline { \mathrm { x } } , \overline { \mathrm { y } } , \overline { \mathrm { z } } ) = \left( 0,0 , \frac { 14 } { 3 } \right)$
C) $( \bar { x } , \bar { y } , \bar { z } ) = \left( 0,0 , \frac { 49 } { 12 } \right)$
D) $( \bar { x } , \bar { y } , \bar { z } ) = \left( 0,0 , \frac { 35 } { 8 } \right)$

Question

Quizplus · Accepted Answer

The symmetry of the solid and the density function about the z-axis implies that $\bar{x} = 0$ and $\bar{y} = 0$. To find $\bar{z}$, we use the fact that the density is proportional to the distance from the plane $z = 7$, which means the density function can be represented as $\rho(z) = k(7 - z)$ for some constant $k$. The center of mass $\bar{z}$ can be found by integrating over the volume of the solid, taking into account the varying density. The calculation involves setting up the integral for the moment about the xy-plane and dividing by the total mass of the solid. The correct calculation leads to $\bar{z} = \frac{21}{5}$.

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Solve the Problem $z = \frac { 7 } { 8 } \sqrt { x ^ { 2 } + y ^ { 2 } }$