Let's address each part step by step: --- ### **Given Curve:** \[ \mathbf{r}(t) = -\frac{t}{2} \cos(\pi t)\mathbf{i} + t^2\mathbf{j} + \frac{2t}{3} \sin(\pi t)\mathbf{k}, \quad t \in [0, 4] \] --- ## **(a) 2D Projection & 3D Drawing** **2D Projection (in the xz-plane):** The xz-projection of the curve uses only the \( x \) and \( z \) components: - \( x(t) = -\frac{t}{2}\cos(\pi t) \) - \( z(t) = \frac{2t}{3}\sin(\pi t) \) **3D Drawing:** All components: - \( x(t) = -\frac{t}{2}\cos(\pi t) \) - \( y(t) = t^2 \) - \( z(t) = \frac{2t}{3}\sin(\pi t) \) You can plot these using Python (matplotlib), Mathematica, or by hand. For this text solution, I'll describe the steps: **How to Plot** - For 2D: Plot \( x(t) \) vs \( z(t) \) - For 3D: Plot \( (x(t), y(t), z(t)) \) in 3D space for \( t \) from 0 to 4.5 --- ## **(b) Find the Velocity and Speed** ### **Velocity:** \[ \mathbf{v}(t) = \frac{d\mathbf{r}}{dt} \] Differentiate each component: - \( x(t) = -\frac{t}{2}\cos(\pi t) \) - \( x'(t) = -\frac{1}{2}\cos(\pi t) + \frac{t}{2}\pi \sin(\pi t) \) - \( y(t) = t^2 \) - \( y'(t) = 2t \) - \( z(t) = \frac{2t}{3}\sin(\pi t) \) - \( z'(t) = \frac{2}{3}\sin(\pi t) + \frac{2t}{3} \pi \cos(\pi t) \) So, \[ \mathbf{v}(t) = \left[ -\frac{1}{2}\cos(\pi t) + \frac{t}{2}\pi \sin(\pi t) \right]\mathbf{i} + 2t\mathbf{j} + \left[ \frac{2}{3}\sin(\pi t) + \frac{2t}{3}\pi \cos(\pi t) \right]\mathbf{k} \] --- ### **Speed:** \[ s(t) = |\mathbf{v}(t)| \] \[ s(t) = \sqrt{ \left( -\frac{1}{2}\cos(\pi t) + \frac{t}{2}\pi \sin(\pi t) \right)^2 + (2t)^2 + \left( \frac{2}{3}\sin(\pi t) + \frac{2t}{3}\pi \cos(\pi t) \right)^2 } \] --- ## **(c) Arc Length of the Curve** Arc length from \( t=a \) to \( t=b \) is: \[ L = \int_{a}^{b} |\mathbf{v}(t)| \, dt \] So, \[ L = \int_{0}^{4} s(t) \, dt \] **You can use numerical integration (e.g., Simpson's Rule or with a calculator) to evaluate this integral.** --- ### **Estimation and Visual Bound** - The 3D plot from (a) gives a visual sense of the curve's length—trace the curve and estimate a lower bound by summing straight-line segments between points. - The actual arc length (from the integral) will be slightly greater than this lower bound. --- ## **Summary Table** | Part | Expression/Formula | Value/Method | |------|-------------------|--------------| | (a) | See above for 2D/3D plotting | Plot for \( t \in [0, 4.5] \) | | (b) | \( \mathbf{v}(t) \) as derived; \( s(t) = |\mathbf{v}(t)| \) | See above | | (c) | \( L = \int_0^4 s(t)\,dt \) | Numerical integration | --- ### **(Optional) Python Code for Arc Length Calculation** ```python import numpy as np from scipy.integrate import quad def s(t): x = -0.5*t*np.cos(np.pi*t) y = t**2 z = (2*t/3)*np.sin(np.pi*t) dx = -0.5*np.cos(np.pi*t) + 0.5*t*np.pi*np.sin(np.pi*t) dy = 2*t dz = (2/3)*np.sin(np.pi*t) + (2*t/3)*np.pi*np.cos(np.pi*t) return np.sqrt(dx**2 + dy**2 + dz**2) arc_length, _ = quad(s, 0, 4) print("Arc length:", arc_length) ``` --- ### **Final Answers** - (a) See descriptions for 2D/3D plotting. - (b) Velocity and speed are given above. - (c) Arc length is \( L = \int_0^4 s(t)\,dt \), to be evaluated numerically. If you want the exact numeric value for the arc length or a code plot, let me know!

Certainly! Here's a step-by-step guide on how you can generate the graphs for the curve's projection in the xz-plane and the 3D plot for \( t \in [0, 4.5] \). I'll provide sample Python code using `matplotlib` to create these visuals. --- ### **1. 2D Projection in the xz-plane** **Equations:** \[ x(t) = -\frac{t}{2} \cos(\pi t), \quad z(t) = \frac{2t}{3} \sin(\pi t) \] **Python code to plot the projection:** ```python import numpy as np import matplotlib.pyplot as plt # Define the parameter t t = np.linspace(0, 4.5, 500) # from 0 to 4.5 # Calculate x(t) and z(t) x = - (t / 2) * np.cos(np.pi * t) z = (2 * t / 3) * np.sin(np.pi * t) # Plot xz-plane projection plt.figure(figsize=(8, 6)) plt.plot(x, z, label='Projection in xz-plane') plt.xlabel('x(t)') plt.ylabel('z(t)') plt.title('Projection of the Curve in the xz-plane') plt.grid(True) plt.legend() plt.show() ``` --- ### **2. 3D Plot of the Curve** **Equations:** \[ x(t) = -\frac{t}{2}\cos(\pi t), \quad y(t) = t^2, \quad z(t) = \frac{2t}{3}\sin(\pi t) \] **Python code for 3D visualization:** ```python from mpl_toolkits.mplot3d import Axes3D # Calculate y(t) y = t**2 # Plot in 3D fig = plt.figure(figsize=(10, 8)) ax = fig.add_subplot(111, projection='3d') ax.plot(x, y, z, color='b', label='3D Curve') ax.set_xlabel('x(t)') ax.set_ylabel('y(t)') ax.set_zlabel('z(t)') ax.set_title('3D Plot of the Particle Trajectory') ax.legend() plt.show() ``` --- ### **Summary:** - The **first graph** shows how the particle's x and z coordinates evolve, giving insight into the curve's shape in the xz-plane. - The **second graph** provides a full 3D visualization, illustrating the particle's motion in space. --- ### **Note:** You can run these scripts in a Python environment with `numpy` and `matplotlib` installed (e.g., Jupyter Notebook, Anaconda, or any Python IDE). If you'd like, I can generate these plots for you and share the images directly.