Given the parametric equations $x=cos\left(8t\right)$, $y=sin\left(8t\right)$, $z=e^{0.8t}$, $t\ge 0$, which of the following best describes the graph of these equations?

• A. A shrinking spiral that approaches the origin as $t$ increases
• B. A spiral that winds upward and outward along the z-axis, forming an expanding helix as $t$ increases
• C. A circle of constant radius in the xy-plane at $z=1$
• D. A straight line moving diagonally through space

Answer: (B)

Verified step by step guidance

1. Identify the parametric equations given: \(x = \cos(8t)\), \(y = \sin(8t)\), and \(z = e^{0.8t}\), with \(t \geq 0\).
2. Analyze the \(x\) and \(y\) components: since \(x = \cos(8t)\) and \(y = \sin(8t)\), these represent a point moving around a circle of radius 1 in the \(xy\)-plane, because \(\cos^2(8t) + \sin^2(8t) = 1\).
3. Examine the \(z\) component: \(z = e^{0.8t}\) is an exponential function that increases as \(t\) increases, meaning the point moves upward along the \(z\)-axis and the height grows exponentially.
4. Combine the \(xy\) and \(z\) components: the point moves in a circular path in the \(xy\)-plane while simultaneously moving upward exponentially in \(z\), creating a spiral that expands upward.
5. Conclude the shape: since the radius in the \(xy\)-plane remains constant but the \(z\) value increases exponentially, the graph forms an expanding helix winding upward along the \(z\)-axis.