Let \(a, b, c\) be the lengths of the sides of a triangle and let \(\theta\) be the angle opposite the side of length \(c\). Find the differential do and approximate c when \(\mathrm{a}=6.20, \mathrm{~b}=5.90\), and \(0=58^{\circ}\).

Evaluate a) $\left.^{2} \int_{0}\left[\mathrm{dx} / \sqrt{\\{}\left(4-\mathrm{x}^{2}\right)\left(9-\mathrm{x}^{2}\right)\right\\}\right]$ \(\quad\) b) $\left.{ }^{\infty} \mathrm{I}_{1}\left[\mathrm{du} / \sqrt{\\{}\left(\mathrm{u}^{2}-1\right)\left(\mathrm{u}^{2}+3\right)\right\\}\right]$ in terms of elliptic integrals.

Find the dimensions of the box of largest volume which can be fitted inside the ellipsoid $$ \left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right)=1 $$ assuming that each edge of the box is parallel to a coordinate axis.

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?

Derive the Equation of Continuity for fluid flows: $$ (\partial p / \partial t)=-\operatorname{div} \rho \mathrm{V}^{-} $$ where \(\rho(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t})\) and \(\mathrm{V}^{-}(\mathrm{x}, \mathrm{y}, z, \mathrm{t})\) are, respectively, the fluid density and velocity at the point \((x, y, z)\) at time t. Conclude \(\operatorname{div} \mathrm{V}^{-}=0\) if the fluid is incompressible.

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 \(t=4\)