Given a stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec.

To find the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing

Let r be the radius of the circle and A be the area of the circle

When stone is dropped into the lake waves moves in circle at speed of 4cm/sec. i.e., radius of the circle increases at a rate of 4cm/sec

\( \dfrac{dr}{dt}\) = 4cm/sec ...(i)

As we know that the area of the circle is πr^{2}

\(\dfrac{dA}{dt}=2\pi r\times 4\)

Therefore, when the radius of the circular wave is 10 cm,

the above equation becomes \( \dfrac{dA}{dt}\) = 2π x 10 x 4

\( \dfrac{dA}{dt}\)= 80 πcm^{2}/sec

Thus, the enclosed area is increasing at the rate of 80 πcm^{2}/sec.

Given the sum of the perimeters of a square and a circle, show that the sum of their areas is least when one side of the square is equal to diameter of the circle.

A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the wire should be cut so that the sum of the areas of the square and triangle is minimum?

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?

Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm^{3}, which has the minimum surface area?

Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.