91Ó°ÊÓ

Find the path of a heat-seeking particle placed at point \(P\) on a metal plate with a temperature field \(T(x, y)\). $$ \begin{array}{ll} \underline{\text { Temperature Field }} & \underline{\text {Point}} \\ T(x, y)=100-x^{2}-2 y^{2} &\quad P(4,3) \end{array} $$

The temperature at the point \((x, y)\) on a metal plate is modeled by \(T(x, y)=400 e^{-\left(x^{2}+y\right) / 2}, \quad x \geq 0, y \geq 0\). (a) Use a computer algebra system to graph the temperature distribution function. (b) Find the directions of no change in heat on the plate from the point \((3,5)\) (c) Find the direction of greatest increase in heat from the point \((3,5)\)

Construction Cost A rectangular box with an open top has a length of \(x\) feet, a width of \(y\) feet, and a height of \(z\) feet. It costs \(\$ 0.75\) per square foot to build the base and \(\$ 0.40\) per square foot to build the sides. Write the cost \(C\) of constructing the box as a function of \(x, y\), and \(z\).

Volume A propane tank is constructed by welding hemispheres to the ends of a right circular cylinder. Write the volume \(V\) of the tank as a function of \(r\) and \(l\), where \(r\) is the radius of the cylinder and hemispheres, and \(I\) is the length of the cylinder.

The temperature at any point \((x, y)\) in a steel plate is \(T=500-0.6 x^{2}-1.5 y^{2}\), where \(x\) and \(y\) are measured in meters. At the point \((2,3)\), find the rate of change of the temperature with respect to the distance moved along the plate in the directions of the \(x\) - and \(y\) -axes.