Use the alternative curvature formula \( \kappa=\frac{|a \times v|}{|v|^{3}} \) to find the curvature of the following parameterized curve. \( r(t)=\langle 5 \cos t, \sqrt{5} \sin t, 2 \cos t\rangle \) \( \kappa=\square \)

To find the curvature \( \kappa \) of the parameterized curve given by \( r(t) = \langle 5 \cos t, \sqrt{5} \sin t, 2 \cos t \rangle \), we will follow these steps: 1. **Calculate the velocity vector \( v(t) \)** by differentiating \( r(t) \) with respect to \( t \). 2. **Calculate the acceleration vector \( a(t) \)** by differentiating \( v(t) \) with respect to \( t \). 3. **Compute the cross product \( a \times v \)**. 4. **Calculate the magnitudes \( |a \times v| \) and \( |v| \)**. 5. **Substitute these values into the curvature formula \( \kappa = \frac{|a \times v|}{|v|^3} \)**. Let's start with step 1 and calculate the velocity vector \( v(t) \): \[ v(t) = \frac{dr}{dt} = \left\langle \frac{d}{dt}(5 \cos t), \frac{d}{dt}(\sqrt{5} \sin t), \frac{d}{dt}(2 \cos t) \right\rangle \] Calculating the derivatives: \[ v(t) = \left\langle -5 \sin t, \sqrt{5} \cos t, -2 \sin t \right\rangle \] Next, we will calculate the acceleration vector \( a(t) \): \[ a(t) = \frac{dv}{dt} = \left\langle \frac{d}{dt}(-5 \sin t), \frac{d}{dt}(\sqrt{5} \cos t), \frac{d}{dt}(-2 \sin t) \right\rangle \] Calculating the derivatives: \[ a(t) = \left\langle -5 \cos t, -\sqrt{5} \sin t, -2 \cos t \right\rangle \] Now, we will compute the cross product \( a \times v \): \[ a \times v = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -5 \cos t & -\sqrt{5} \sin t & -2 \cos t \\ -5 \sin t & \sqrt{5} \cos t & -2 \sin t \end{vmatrix} \] Calculating the determinant: \[ a \times v = \mathbf{i} \left( (-\sqrt{5} \sin t)(-2 \sin t) - (-2 \cos t)(\sqrt{5} \cos t) \right) - \mathbf{j} \left( (-5 \cos t)(-2 \sin t) - (-2 \cos t)(-5 \sin t) \right) + \mathbf{k} \left( (-5 \cos t)(\sqrt{5} \cos t) - (-5 \sin t)(-\sqrt{5} \sin t) \right) \] Calculating each component: 1. For \( \mathbf{i} \): \[ 2\sqrt{5} \sin^2 t - 2\sqrt{5} \cos^2 t = 2\sqrt{5} (\sin^2 t - \cos^2 t) \] 2. For \( \mathbf{j} \): \[ 10 \cos t \sin t - 10 \sin t \cos t = 0 \] 3. For \( \mathbf{k} \): \[ -5\sqrt{5} \cos^2 t + 5\sqrt{5} \sin^2 t = 5\sqrt{5} (\sin^2 t - \cos^2 t) \] Thus, we have: \[ a \times v = \left\langle 2\sqrt{5} (\sin^2 t - \cos^2 t), 0, 5\sqrt{5} (\sin^2 t - \cos^2 t) \right\rangle \] Next, we calculate the magnitude \( |a \times v| \): \[ |a \times v| = \sqrt{(2\sqrt{5} (\sin^2 t - \cos^2 t))^2 + 0^2 + (5\sqrt{5} (\sin^2 t - \cos^2 t))^2} \] Calculating this: \[ |a \times v| = \sqrt{(4 \cdot 5 (\sin^2 t - \cos^2 t)^2) + (25 \cdot 5 (\sin^2 t - \cos^2 t)^2)} = \sqrt{(4 + 25) \cdot 5 (\sin^2 t - \cos^2 t)^2} = \sqrt{145} |\sin^2 t - \cos^2 t| \] Now, we calculate the magnitude \( |v| \): \[ |v| = \sqrt{(-5 \sin t)^2 + (\sqrt{5} \cos t)^2 + (-2 \sin t)^2} = \sqrt{25 \sin^2 t + 5 \cos^2 t + 4 \sin^2 t} = \sqrt{29 \sin^2 t + 5 \cos^2 t} \] Finally, we substitute these values into the curvature formula: \[ \kappa = \frac{|a \times v|}{|v|^3} = \frac{\sqrt{145} |\sin^2 t - \cos^2 t|}{(29 \sin^2 t + 5 \cos^2 t)^{3/2}} \] Thus, the curvature \( \kappa \) is: \[ \kappa = \frac{\sqrt{145} |\sin^2 t - \cos^2 t|}{(29 \sin^2 t + 5 \cos^2 t)^{3/2}} \]