EXERCISE 8(d)

A cylindrical vessel of capacity $500$ cubic metres open at the top is to be constructed. Find the dimensions of the vessel if the material used is minimum given that the thickness of the material used is $2 \mathrm{cm}$.

Find the coordinates of the point on the curve $x^{2}y-x+y=0$ where the slope of the tangent is maximum.

Find the points on the curve $y=x^2+1$ which are nearest to the point $(0,2)$.

Show that the minimum distance of a point on the curve $\frac{a^2}{x^2}+\frac{b^2}{y^2}=1$ from the origin is $a+b$.

Show that the vertical angle of a right circular cone of minimum curved surface that circumscribes a given sphere is $2{\sin }^{-1}(\sqrt{2}-1)$.

Show that the semi-vertical angle of a right circular cone of maximum volume that circumscribes a given sphere is ${\sin }^{-1}\left(\frac{1}{3}\right)$.

Show that the shortest distance of the point $(0,8a)$ from the curve $a{x}^2=y^3$ is $2a\sqrt{11}$.

Show that the triangle of greatest area that can be inscribed in a circle is equilateral.