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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Preview Activity 13.9.1 .
Recall the standard parameterization of the unit circle that is given by
\begin{equation*}
x(t) = \cos(t) \ \ \ \ \text{ and } \ \ \ \ y(t) = \sin(t),
\end{equation*}
where \(0 \le t \le 2\pi\text{.}\)
Determine a parameterization of the circle of radius 1 in
\(\R^3\) that has its center at
\((0,0,1)\) and lies in the plane
\(z=1\text{.}\)
Determine a parameterization of the circle of radius 1 in 3-space that has its center at
\((0,0,-1)\) and lies in the plane
\(z=-1\text{.}\)
Determine a parameterization of the circle of radius 1 in 3-space that has its center at
\((0,0,5)\) and lies in the plane
\(z=5\text{.}\)
Taking into account your responses in (a), (b), and (c), describe the graph that results from the set of parametric equations
\begin{equation*}
x(s,t) = \cos(t), \ \ \ \ y(s,t) = \sin(t), \ \ \ \ \text{ and } \ \ \ \ z(s,t) = s,
\end{equation*}
where \(0 \le t \le 2\pi\) and \(-5 \le s \le 5\text{.}\) Explain your thinking.
Just as a cylinder can be viewed as a “stack” of circles of constant radius, a cone can be viewed as a stack of circles with varying radius. Modify the parametrizations of the circles above in order to construct the parameterization of a cone whose vertex lies at the origin, whose base radius is 4, and whose height is 3, where the base of the cone lies in the plane
\(z = 3\text{.}\) Use appropriate technology to plot the parametric equations you develop. (Hint: The cross sections parallel to the
\(xy\) -plane are circles, with the radii varying linearly as
\(z\) increases.)
