Find \( d w / d t \) using the appropriate Chain Rule. \[ \begin{array}{l}w=x^{2}+y^{2} \\ x=4 t, y=7 t\end{array} \]

To find \( \frac{dw}{dt} \) using the Chain Rule, we start by applying the derivative of \( w \) with respect to \( t \) through its dependence on \( x \) and \( y \): 1. We first find \( \frac{dw}{dx} \) and \( \frac{dw}{dy} \): \[ w = x^2 + y^2 \implies \frac{dw}{dx} = 2x \quad \text{and} \quad \frac{dw}{dy} = 2y. \] 2. Next, we find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \): \[ x = 4t \implies \frac{dx}{dt} = 4, \quad y = 7t \implies \frac{dy}{dt} = 7. \] 3. Now, we can use the Chain Rule: \[ \frac{dw}{dt} = \frac{dw}{dx} \cdot \frac{dx}{dt} + \frac{dw}{dy} \cdot \frac{dy}{dt}. \] 4. Substituting the derivatives: \[ \frac{dw}{dt} = (2x) \cdot (4) + (2y) \cdot (7). \] 5. Substitute \( x = 4t \) and \( y = 7t \) into the expression: \[ \frac{dw}{dt} = 2(4t) \cdot 4 + 2(7t) \cdot 7. \] 6. Simplifying this gives: \[ \frac{dw}{dt} = 32t + 98t = 130t. \] Therefore, \[ \frac{dw}{dt} = 130t. \]