Answer
The parametric equations describe a curve that spirals upwards. As \(t\) increases, the curve spirals around the cone with increasing radius.
Work Step by Step
\[
x=t\cos(t), \quad y=t\sin(t), \quad z=t
\]
We need to show that this curve lies on the cone defined by the equation:
\[
x^2+y^2=z^2
\]
Calculate \(x^2+y^2\):
\[
x^2+y^2=(t\cos(t)+t\sin(t))^2 \\
x^2+y^2=t^2(\cos^2(t)+\sin^2(t))
\]
Since \(\cos^2(t)+\sin^2(t)=1\)
\[x^2+y^2=t^2\]
Since \(z=t\), we have:
\[
z^2=t^2\\
x^2+y^2=t^2 \quad and \quad z^2=t^2 \\
\]
So we have:
\[
x^2+y^2=z^2
\]
