Question

Give a parametric description for a cone with radius $a$ and height $h,$ including the intervals for the parameters.

Short answer

Answer: The parametric description of a cone with radius a and height h is given by: $(x,y,z)=(a(1-t)\cos (s),a(1-t)\sin (s),ht)$ The intervals for the parameters are $0\le t\le 1$ and $0\le s\le 2\pi$.

Step-by-step solution

Write the parameterized equation for the cone

We need to find a parametric equation that describes the cone, in terms of coordinates $(x,y,z)$. For a cone with radius $a$ and height $h$, we can use the parameters $t$ and $s$ to define the coordinates as follows: $(x,y,z)=(a(1-t)\cos (s),a(1-t)\sin (s),ht)$

Determine the intervals for the parameters

Now that we have the parameterized equation, we need to define the intervals for the parameters, $t$ and $s$. To describe the entire cone, we need to take the following intervals for the parameters: For $t$: Since $t$ defines the height of the cone, the interval for $t$ should be $0\le t\le 1$. For $s$: Since $s$ defines the circular base of the cone, the interval for $s$ should be $0\le s\le 2\pi$.

Final Solution

The parametric description of a cone with radius $a$ and height $h$ is given by: $(x,y,z)=(a(1-t)\cos (s),a(1-t)\sin (s),ht)$ with the intervals for the parameters being $0\le t\le 1$ and $0\le s\le 2\pi$.