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

248 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.

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 iswhere $$\scriptstyle{x}$$ 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.&emsp;$\scriptstyle{ax=by=cz=\sqrt[3]{\frac{abcg}{n}}}$.|undefined