Let S be the oriented upwards. Use a computer surface $ z=100-x^{2}-y^{2}, z \geq 0 $ algebra system to find the rate of mass flow of a fluid of density $\rho$ through $S$ if the velocity field is given by $\mathbf { F } ( x , y , z ) = 0.5 z k$.
A) $25,000 \pi \rho$
B) $5,000 \pi \rho$
C) $25 \pi \rho$
D) $250,000 \pi \rho$
E) $2,500 \pi \rho$

Question

Quizplus · Accepted Answer

The rate of mass flow through the surface $S$ is given by the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$, where $\mathbf{F}$ is the velocity field and $d\mathbf{S}$ is the oriented surface element. Given $\mathbf{F} = 0.5z\mathbf{k}$, the mass flow rate is proportional to the integral of $z$ over the surface, which is a paraboloid. The equation $z = 100 - x^2 - y^2$ describes a paraboloid opening downwards with a vertex at $(0,0,100)$ and intersecting the $z=0$ plane in a circle of radius $10$. The mass flow rate can be calculated using cylindrical coordinates, integrating over the disk of radius $10$ in the $xy$-plane, and taking into account the density $\rho$. The correct calculation leads to the result $2,500 \pi \rho$, indicating the rate of mass flow through the surface.

Let S Be the Oriented Upwards z=100−x2−y2,z≥0 z=100-x^{2}-y^{2}, z \geq 0 z=100−x2−y2,z≥0

Let S Be the Oriented Upwards $z=100-x^{2}-y^{2}, z \geq 0$