Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/272

 6. Show that the surface of a rectangular parallelepiped of given volume is least when the solid is a cube.

 7. Examine $\scriptstyle {x^{4}+y^{4}-x^{2}+xy-y^{2}}$ for maximum and minimum values.

Ans.	Maximum when $\scriptstyle {x=0,\ y=0}$;
	minimum when $\scriptstyle {x=y\pm {\frac {1}{2}}}$, and when $\scriptstyle {x=-y=\pm {\frac {1}{2}}{\sqrt {3}}}$.

 8. Show that when the radius of the base equals the depth, a steel cylindrical standpipe of a given capacity requires the least amount of material in its construction.

 9. Show that the most economical dimensions for a rectangular tank to hold a given volume are a square base and a depth equal to one half the side of the base.

10. The electric time constant of a cylindrical coil of wire is

$\scriptstyle {u={\frac {mxyz}{ax+by+cz}}}$,

where $\scriptstyle {x}$ is the mean radius, $\scriptstyle {y}$ is the difference between the internal and external radii, $\scriptstyle {z}$ is the axial length, and $\scriptstyle {m,\ a,\ b,\ c}$ are known constants. The volume of the coil is $\scriptstyle {nxyz=g}$. Find the values of $\scriptstyle {x,\ y,\ z}$ which make $\scriptstyle {u}$ a minimum if the volume of the coil is fixed.

Ans. $\scriptstyle {ax=by=cz={\sqrt[{3}]{\frac {abcg}{n}}}}$.