Let's break down the problem step by step: --- ### **Given** A particle moves along a path given by: \[ \vec{r}(t) = \langle 4\cos t,\, 4\sin t,\, 3t \rangle \] with \( t \in [0,4\pi] \) or \( t \in [-2\pi,2\pi] \). We need to **sketch the curve**. --- ### **Step 1: Understanding the Path** - The \(x\) and \(y\) components are: \[ x(t) = 4\cos t,\quad y(t) = 4\sin t \] These describe a circle of **radius 4** in the \(xy\)-plane as \(t\) goes from \(0\) to \(2\pi\). - The \(z\) component is: \[ z(t) = 3t \] As \(t\) increases, \(z\) increases linearly. **This is a helix:** The particle moves in a circle of radius 4, while moving upward (or downward) with \(z = 3t\). --- ### **Step 2: Parameter Interval** - If \( t \in [0, 4\pi] \): - \( x \) and \( y \) complete **two full circles** (since one circle for \(0\) to \(2\pi\)). - \( z \) goes from \(0\) to \(3(4\pi) = 12\pi\). - If \( t \in [-2\pi, 2\pi] \): - Also covers two full circles, but \(z\) goes from \(-6\pi\) to \(6\pi\). --- ### **Step 3: Sketching the Curve** #### **1. Projection onto the \(xy\)-plane:** - The curve projects to a circle of radius 4. #### **2. Helix in 3D:** - As \(t\) increases, the point moves counterclockwise around the circle (since \(\cos t\), \(\sin t\)), but at the same time, \(z\) increases linearly, forming a **helix** that wraps around the cylinder \(x^2 + y^2 = 16\). #### **3. Key Points:** - At \(t = 0\): \((4, 0, 0)\) - At \(t = \pi/2\): \((0, 4, 3\pi/2)\) - At \(t = \pi\): \((-4, 0, 3\pi)\) - At \(t = 3\pi/2\): \((0, -4, 9\pi/2)\) - At \(t = 2\pi\): \((4, 0, 6\pi)\) — one full turn - At \(t = 4\pi\): \((4, 0, 12\pi)\) — two full turns --- ### **Step 4: Final Answer (Description and Sketch)** #### **Final Answer:** The particle traces a helix of radius 4 centered on the \(z\)-axis, with a vertical spacing of \(3 \cdot 2\pi = 6\pi\) per turn. For \(t \in [0, 4\pi]\), the helix starts at \((4,0,0)\) and finishes at \((4,0,12\pi)\) after two full turns. #### **Sketch (Description):** 1. Draw the \(z\)-axis vertically. 2. Draw a cylinder of radius 4 about the \(z\)-axis. 3. Mark the starting point at \((4,0,0)\). 4. Draw a spiral (helix) that wraps counterclockwise (when viewed from above), rising as \(z\) increases, making **two full turns** by \(z = 12\pi\). 5. The projection onto the \(xy\)-plane is a circle. --- **In summary**: The path is a helix around the cylinder \(x^2 + y^2 = 16\) with pitch \(6\pi\) (vertical distance per turn), making two turns as \(z\) goes from \(0\) to \(12\pi\). --- #### **Optional: Parametric Plot Example (for visualization)** If you want to plot this in, say, Python (matplotlib): ```python import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D t = np.linspace(0, 4*np.pi, 500) x = 4 * np.cos(t) y = 4 * np.sin(t) z = 3 * t fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.plot(x, y, z) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel('z') plt.show() ``` --- Let me know if you would like the sketch itself!

Certainly! Let's go through a detailed step-by-step solution, including the calculations, to understand and sketch the given curve. --- ### **Given:** The particle moves along the path parameterized by: \[ \vec{r}(t) = \langle 4\cos t,\, 4\sin t,\, 3t \rangle \] with \( t \in [0, 4\pi] \). --- ### **Step 1: Understand the components of the motion** - **\(x(t) = 4\cos t\)** - **\(y(t) = 4\sin t\)** - **\(z(t) = 3t\)** --- ### **Step 2: Describe the projection onto the \(xy\)-plane** The \(x\) and \(y\) components define a circle: \[ x^2 + y^2 = (4\cos t)^2 + (4\sin t)^2 = 16 \cos^2 t + 16 \sin^2 t = 16 (\cos^2 t + \sin^2 t) = 16 \] **This is a circle of radius 4** centered at the origin in the \(xy\)-plane. --- ### **Step 3: Describe the \(z\)-component** The height varies linearly: \[ z(t) = 3t \] - When \(t=0\), \(z=0\). - When \(t=4\pi\), \[ z = 3 \times 4\pi = 12\pi \] which is approximately \(12 \times 3.1416 \approx 37.699\). --- ### **Step 4: Understanding the motion** - As \(t\) increases from \(0\) to \(4\pi\), the particle completes **two full revolutions** around the circle (since one full revolution occurs over \(t\) from 0 to \(2\pi\)). - The particle moves **upward** along the \(z\)-axis from 0 to \(12\pi \approx 37.7\). - The path traces a **helix** wrapped around a cylinder of radius 4. --- ### **Step 5: Key points** Let's find some specific points on the path: | \(t\) | \(x(t)\) | \(y(t)\) | \(z(t)\) | Description | |--------|-----------|-----------|--------|--------------| | 0 | \(4 \cos 0=4\) | \(4 \sin 0=0\) | \(0\) | Starting point at \((4,0,0)\) | | \(\pi/2\) | \(4 \cos \pi/2=0\) | \(4 \sin \pi/2=4\) | \(3 \times \pi/2 \approx 4.712\) | Quarter turn, top point at \((0,4,4.712)\) | | \(\pi\) | \(-4\) | 0 | \(3 \pi \approx 9.425\) | Half turn, at \((-4,0,9.425)\) | | \(3\pi/2\) | 0 | \(-4\) | \(3 \times 3\pi/2 \approx 14.137\) | Three-quarters turn, at \((0,-4,14.137)\) | | \(2\pi\) | \(4\) | 0 | \(6 \pi \approx 18.85\) | Full turn, back to \((4,0,18.85)\) | Similarly, for \(t=4\pi\): \[ x=4 \cos 4\pi=4,\quad y=4 \sin 4\pi=0,\quad z=3 \times 4\pi=12\pi \approx 37.699 \] which indicates the particle completes **two full revolutions** in the \(xy\)-plane, rising from \(z=0\) to \(z \approx 37.7\). --- ### **Step 6: Sketching the curve** - **Projection onto the \(xy\)-plane:** circle of radius 4 centered at the origin. - **Vertical motion:** linear increase in \(z\) from 0 to \(12\pi\). - **Path shape:** a **helix** wrapping around the cylinder of radius 4, with **two turns**. --- ### **Final summary:** - **The path is a helix** with: - **Radius:** 4 - **Number of turns:** 2 (from \(t=0\) to \(t=4\pi\)) - **Vertical rise per turn:** \( \frac{12\pi}{2} = 6\pi \approx 18.85 \) - **Equation of the curve:** \[ x=4\cos t,\quad y=4\sin t,\quad z=3t,\quad t \in [0, 4\pi] \] --- ### **Visual sketch instructions:** 1. Draw the \(xy\)-plane with a circle of radius 4. 2. Mark the starting point at \((4,0,0)\). 3. Sketch a spiral wrapping around the cylinder, ascending linearly, completing two full revolutions from \(z=0\) to approximately 37.7. 4. Indicate the height increases linearly with \(t\). --- **This detailed step-by-step process should help you understand and visualize the curve!**