Given the parametric equations $x=t\cos \left(t\right)$, $y=t$, $z=t\sin \left(t\right)$ for $t\ge 0$, which of the following best describes the graph of these equations?

• A. A straight line passing through the origin in three-dimensional space
• B. A paraboloid opening along the $y$-axis
• C. A circle in the $x,z$ plane at $y=0$
• D. A spiral (helix) that winds upward along the $y$-axis

Answer: (D)

Verified step by step guidance

1. Identify the parametric equations given: \(x = t \cdot \cos(t)\), \(y = t\), and \(z = t \cdot \sin(t)\), where \(t \geq 0\).
2. Observe that \(y = t\) increases linearly with the parameter \(t\), which means the curve moves upward along the \(y\)-axis as \(t\) increases.
3. Focus on the \(x\) and \(z\) components: \(x = t \cos(t)\) and \(z = t \sin(t)\). These resemble the parametric form of a spiral in the \(x\)-\(z\) plane, where the radius from the origin increases proportionally to \(t\).
4. Recall that the parametric equations \(x = r \cos(\theta)\) and \(z = r \sin(\theta)\) describe a circle in the \(x\)-\(z\) plane with radius \(r\). Here, the radius \(r = t\) grows with \(t\), so the curve spirals outward as it moves upward along \(y\).
5. Combine these observations: as \(t\) increases, the point moves upward along \(y\) while simultaneously spiraling outward in the \(x\)-\(z\) plane, forming a three-dimensional spiral or helix winding upward along the \(y\)-axis.