Tamil Nadu Board Solutions for Chapter: Applications of Differential Calculus, Exercise 8: Exercise 8

Find the smallest possible value of $x^2+y^2$ given that $x+y=10$.

A garden is to be laid out in a rectangular area and protected by wire fence. What is the largest possible area of the fenced garden with $40$ metres of wire in square meters.

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain $1,80,000\mathrm{sq}.\mathrm{mtrs}$ in order to provide enough grass for herds. No fencing is needed along the river. What is the length of the minimum needed fencing material in meters?

Prove that among all the rectangles of the given perimeter, the square has the maximum area.

Find the dimensions of the largest rectangle that can be inscribed in a semi circle of radius $r\mathrm{cm}$.

A manufacturer wants to design an open box having a square base and a surface area of $108\mathrm{sq}.\mathrm{cm}$. Determine the dimensions of the box for the maximum volume.

The volume of a cylinder is given by the formula $V=\pi r^{2}h$. Find the greatest and least values of $V$ if $r+h=6.$

A hollow cone with base radius $r$ and height $h$ is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is $\frac{4}{9}$ times volume of the cone.