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? \( \frac{\partial \rho}{\partial y}=x \) (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{d y}{d t}=4 \) cos \( t \) (Type an expression using \( t \) as the variable.) a. What is the rate of change of the density with respect to \( t \) on the edge of the plate? \( \rho^{\prime}(t)=16 \) cos \( 2 t \) (Type an expression using \( t \) as the variable.) b. At what point(s) on the edge of the plate is the density a maximum? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O. AThe density on the edgetyf the plate has a maximum value at the point(s) \( \square \). (Type an ordered pair. Type an exact answer. Use a comma to separate answers as needed.) B. The density on the edge of the plate has no maximum value.
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Bonus Knowledge
Did you know that in ancient Greece, the study of circular shapes and their dimensions was quite advanced? Mathematicians like Archimedes were exploring the properties of circles long before calculus came about. The circular plate in your problem brings to mind their work, especially in understanding areas and volumes, which would later inspire the mathematical tools used today in physics and engineering to analyze objects' densities! In the real world, understanding density and its rates of change can be crucial for engineers designing structures or materials. For instance, in the aerospace industry, knowing how density varies can impact the performance of different components. Such concepts are essential in optimizing aircraft design for weight distribution and structural integrity, ensuring that planes fly smoothly and safely through the skies!