Show how to find the semiaxes of the ellipse in which the plane $$ \begin{aligned} \mathrm{P}(\mathrm{x}, \mathrm{y}, z) &=\mathrm{pz}+\mathrm{qy}+\mathrm{rz} \\\ &=0(\mathrm{pqr} \neq 0) \end{aligned} $$ cuts the ellipsoid $E(x, y, z)=\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right)$ \(=1 .(0

A particle slides freely in a tube which rotates in a vertical plane about its midpoint with constant angular velocity w. If \(\mathrm{x}\) is the distance of the particle from the midpoint of the tube at time \(t\) and if the tube is horizontal with \(t=0\), show that the motion of the particle along the tube is given by $$ \left(d^{2} x / d t^{2}\right)-w^{2} x=-g \text { sin wt. } $$ Solve this equation if $\mathrm{x}=\mathrm{x}_{0}, \mathrm{dx} / \mathrm{dt}=\mathrm{v}_{0}\( when \)\mathrm{t}=0$. For what values of \(\mathrm{x}_{0}\) and \(\mathrm{v}_{0}\) is the motion simple harmonic?

Suppose a student has \(\$ 90\) with which to buy lecture notes at \(\$ 3\) each and packs of beer, at \(\$ 5\) each. Let the function $$ f(x, y)=x y $$ (1) express how much satisfaction is derived from buying \(\mathrm{x}\) lecture notes and \(\mathrm{y}\) packs of beer. What \(\mathrm{x}\) and \(\mathrm{y}\) will maximize the student's pleasure?

The electrostatic field produced by a unit positive charge at 0 is $$ \mathrm{E}=\left(1 / \mathrm{r}^{3}\right) \mathrm{A}^{-} $$ where \(\mid \mathrm{A} \|=\mathrm{r}\). Find the divergence of this field (wherever the field is defined, i.e. at all points except 0).

Which 3 -dimensional rectangular box of a given volume \(\mathrm{V}\) has the least surface area?

A particle moves in the plane according to $$ x=64 \sqrt{3} t, \quad y=64 t-16 t^{2} $$ and is acted on by a force \(\mathrm{F}\) which is directly proportional to the velocity but opposite in direction. Find the work done by \(F\) from \(t=0\) to \(t=4\)