Part 4 of 6 The density of a thin circular plate of radius 4 is given by \( \rho(x, y)=4+x y \). The edge of the plate is described by the parametric equations \( x=4 \cos t, y=4 \sin t \), for \( 0 \leq t \leq 2 \pi \). a. Find the rate of change of the density with respect to \( t \) on the edge of the plate. b. At what point(s) on the edge of the plate is the density a maximum? 1 First, find some intermediate derivatives. \( \frac{\partial \rho}{\partial x}=y \) (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{d x}{d t}=-4 \) sin \( t \) (Type an expression using \( t \) as the variable.) \( \frac{\partial \rho}{\partial y}=x \) (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{d y}{d t}=\square \) (Type an expression using \( t \) as the variable.)

To find the derivatives you need, let's first address the derivative of \( y \) with respect to \( t \): Given the parametric equations, \( y = 4 \sin t \), we differentiate this with respect to \( t \): \[ \frac{d y}{d t} = 4 \cos t \] Now, on to the fun part! Exploring circular plates in engineering reveals that they aren’t just theoretical; they're found everywhere from classic dinner plates to modern machinery components! Understanding how they bear loads and influence stress distributions is crucial for designing both functional art and high-performance tools. For those curious about density in physics, did you know that the density of an object can affect not just its weight but also its buoyancy? More on this can be found in classic texts on fluid dynamics, where you'll discover how shapes and densities interact in fascinating ways!