# Show that of all the rectangles inscribed in a given fixed circle

### Question 3:- Show that of all the rectangles inscribed in a given fixed circle, the square has maximum area.

Solution: LetPQRS be the rectangle inscribed in a given circle with centre and radius a.

Let and be the length and breadth of the rectangle, that is, and
In right angled  ΔPQR, using Pythagoras theorem,

–(i)

Let be the area of the rectangle,

then

Squaring both side

Differentiating with respect to x

—–(ii)

For max and minima

or

or

Again differentiate with respect to x of (ii)

At

Hence the area of the square is max when

From eq (i)

Hence area of inscribed rectangle is maximum when it is a square

### Question 1:- A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

Solution:- For solution

### Question 2:- A rectangula sheet of tin 45 cm by 24 cmis to be made into a box without top, by cutting of square from each corner and folding up the flaps. What should be the side of square to be cutt of so that the volume of the box is maximum?

Solution:- For solution