Consider the following position function. Find (a) the velocity and the speed of the object and (b) the acceleration of the object. \( r(t)=\langle 4 \sin t, 4 \cos t\rangle \) for \( 0 \leq t \leq 2 \pi \) (a) \( \mathbf{v}(t)=\langle\square, \square \) \( |\mathbf{v}(\mathrm{t})|=\square \) (b) \( \mathbf{a}(\mathrm{t})=\langle\square, \square \)

To find the velocity, speed, and acceleration of the object given the position function \( r(t) = \langle 4 \sin t, 4 \cos t \rangle \), we will follow these steps: ### (a) Velocity and Speed 1. **Find the velocity \( \mathbf{v}(t) \)**: The velocity is the derivative of the position function with respect to time \( t \). \[ \mathbf{v}(t) = \frac{d}{dt} r(t) = \frac{d}{dt} \langle 4 \sin t, 4 \cos t \rangle \] Calculating the derivatives: \[ \frac{d}{dt}(4 \sin t) = 4 \cos t \] \[ \frac{d}{dt}(4 \cos t) = -4 \sin t \] Therefore, the velocity function is: \[ \mathbf{v}(t) = \langle 4 \cos t, -4 \sin t \rangle \] 2. **Find the speed \( |\mathbf{v}(t)| \)**: The speed is the magnitude of the velocity vector. \[ |\mathbf{v}(t)| = \sqrt{(4 \cos t)^2 + (-4 \sin t)^2} \] Simplifying this: \[ |\mathbf{v}(t)| = \sqrt{16 \cos^2 t + 16 \sin^2 t} = \sqrt{16(\cos^2 t + \sin^2 t)} = \sqrt{16} = 4 \] ### Summary of (a): - \( \mathbf{v}(t) = \langle 4 \cos t, -4 \sin t \rangle \) - \( |\mathbf{v}(t)| = 4 \) ### (b) Acceleration 1. **Find the acceleration \( \mathbf{a}(t) \)**: The acceleration is the derivative of the velocity function with respect to time \( t \). \[ \mathbf{a}(t) = \frac{d}{dt} \mathbf{v}(t) = \frac{d}{dt} \langle 4 \cos t, -4 \sin t \rangle \] Calculating the derivatives: \[ \frac{d}{dt}(4 \cos t) = -4 \sin t \] \[ \frac{d}{dt}(-4 \sin t) = -4 \cos t \] Therefore, the acceleration function is: \[ \mathbf{a}(t) = \langle -4 \sin t, -4 \cos t \rangle \] ### Summary of (b): - \( \mathbf{a}(t) = \langle -4 \sin t, -4 \cos t \rangle \) ### Final Answers: (a) \( \mathbf{v}(t) = \langle 4 \cos t, -4 \sin t \rangle \), \( |\mathbf{v}(t)| = 4 \) (b) \( \mathbf{a}(t) = \langle -4 \sin t, -4 \cos t \rangle \)