### Finding the Speed at a Given Value of \( t \)In this problem, we are asked to find the speed of a particle at a specific time \( t \). The particle's position as a function of time is given by a vector-valued function \( \mathbf{r}(t) \).#### Given Data:- The position function of the particle is: \[ \mathbf{r}(t) = \langle \sin(3t), \cos(6t), \cos(7t) \rangle. \]- The specific time at which we need to find the speed is \( t = \frac{\pi}{2} \).#### Speed of the Particle:The speed of the particle is the magnitude of its velocity vector \( \mathbf{v}(t) \), which is the derivative of the position function \( \mathbf{r}(t) \).1. **Derivative Calculation**: - First, we find the derivative of each component of \( \mathbf{r}(t) \): \[ \mathbf{v}(t) = \mathbf{r}'(t) = \left\langle \frac{d}{dt}[\sin(3t)], \frac{d}{dt}[\cos(6t)], \frac{d}{dt}[\cos(7t)] \right\rangle. \] - Calculating the derivatives: \[ \mathbf{v}(t) = \langle 3\cos(3t), -6\sin(6t), -7\sin(7t) \rangle. \]2. **Evaluate at \( t = \frac{\pi}{2} \)**: - Substitute \( t = \frac{\pi}{2} \) into \( \mathbf{v}(t) \): \[ \mathbf{v}\left(\frac{\pi}{2}\right) = \left\langle 3\cos\left(3 \cdot \frac{\pi}{2}\right), -6\sin\left(6 \cdot \frac{\pi}{2}\right), -7\sin\left(7 \cdot \frac{\pi}{2}\right) \right\rangle. \] - Simplifying the trigonometric functions: \[ \cos\left(\frac{3

Name: ### Finding the Speed at a Given Value of \( t \)In this problem, we
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: ### Finding the Speed at a Given Value of \( t \)In this problem, we are asked to find the speed of a particle at a specific time \( t \). The particle's position as a function of time is given by a vector-valued function \( \mathbf{r}(t) \).#### Gi