Radius of a variable circle is changing at the rate of 5cm/sec. What is the radius of the circle at the times when area is changing at the rate of 100cm² /sec?

The bottom of a rectangular swimming tank is 50cm \(\times\)20cm.Water is pumped into the tank at the rate of 500c.c/min.Find the rate at which the level of water in the tank rising

A water tank has a shape of an inverted right circular cone with its axis vertical and vertex lowermost. It semi-vertical angle is tan^-1(0.5). Water is poured into it at a constant rate of 5cubic meter per minute. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 10m.

A woman is moving away from a tower 41.5m high at the rate of 2m/sec. Find the rate at which the angle of elevation of the top of the tower is changing, when she is at a distance of 30m from the foot of the tower. Assume that eye level is 1.5m from the ground.

Find the intervals in which the function f given by f(x) =sin x+cos x,0\(\le \)x\(\le \)2\(\pi\),is strightlly increasing or strightly decreasing.

Find the local maxima and local minima, of the function \(f(x)=\sin x-\cos x, 0. Also find the local maximum and local minimum values.

A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 3 mand volume is 75 cm³. If building of tank costs Rs. 100 per square metre for the base and Rs. 50 per square metre for the sides, find the cost of least expensive tank.

A figure consists of a semicircle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum.

Show that semi-vertical angle of a cone of maximum volume and given slant height is \(\cos^{-1}(\frac{1}{\sqrt3})\)

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \(\frac { 4r }{ 3 } \) Also show that the maximum volume of the cone is \(\frac { 8 }{ 27 } \) of the volume of the sphere.