Find an equation of the surface of revolution generated by revolving the curve given below in the indicated coordinate plane about the given axis. $\begin{array}{|c|c|c|}
\hline \text { Equation of Curve } & \text { Coordinate Plane } & \text { Axis of Revolution } \
\hline z=5 x & x z \text {-plane } & z \text {-axis } \
\hline
\end{array}$

A)
$$x ^ { 2 } + y ^ { 2 } = \frac { z } { 5 }$$
B)
$$x ^ { 2 } + z ^ { 2 } = \frac { y ^ { 2 } } { 25 }$$
C)
$$x ^ { 2 } + y ^ { 2 } = \frac { z ^ { 2 } } { 5 }$$
D)
$$x ^ { 2 } + y ^ { 2 } = \frac { z ^ { 2 } } { 25 }$$
E)
$$y ^ { 2 } + z ^ { 2 } = \frac { x ^ { 2 } } { 25 }$$

Question

Find an equation of the surface of revolution generated by revolving the curve given below in the indicated coordinate plane about the given axis. $\begin{array}{|c|c|c|}
\hline \text { Equation of Curve } & \text { Coordinate Plane } & \text { Axis of Revolution } \
\hline z=5 x & x z \text {-plane } & z \text {-axis } \
\hline
\end{array}$

A) 
$$x ^ { 2 } + y ^ { 2 } = \frac { z } { 5 }$$
B)
$$x ^ { 2 } + z ^ { 2 } = \frac { y ^ { 2 } } { 25 }$$
C)
$$x ^ { 2 } + y ^ { 2 } = \frac { z ^ { 2 } } { 5 }$$
D)
$$x ^ { 2 } + y ^ { 2 } = \frac { z ^ { 2 } } { 25 }$$
E)
$$y ^ { 2 } + z ^ { 2 } = \frac { x ^ { 2 } } { 25 }$$

Quizplus · Accepted Answer

The equation $z = 5x$ in the $xz$-plane describes a line. When this line is revolved around the $z$-axis, every point on the line generates a circle in the $xy$-plane with a radius proportional to its $x$-value. The equation of a circle in the $xy$-plane with radius $r$ is $x^2 + y^2 = r^2$. Since $z = 5x$, we substitute $x$ with $\frac{z}{5}$ to get the radius as a function of $z$, leading to $x^2 + y^2 = \left(\frac{z}{5}\right)^2$, which simplifies to $x^2 + y^2 = \frac{z^2}{25}$.

Find an Equation of the Surface of Revolution Generated by Revolving