## 24-25 Largest rectangle inscribed in a circular quadrant

Problem 24
Find the area of the largest rectangle that can be cut from a circular quadrant as in Fig. 76. Problem 25
In Problem 24, draw the graph of A as a function of $\theta$. Indicating the portion of the curve that has a meaning.

## 53 - 55 Solved Problems in Maxima and Minima

Problem 53
Cut the largest possible rectangle from a circular quadrant, as shown in Fig. 40.

Problem 54
A cylindrical tin boiler, open at the top, has a copper bottom. If sheet copper is m times as expensive as tin, per unit area, find the most economical proportions.

Problem 55
Solve Problem 54 above if the boiler is to have a tin cover. Deduce the answer directly from the solution of Problem 54.

## 04 Four overlapping semi-circles inside a square

Example 04
The figure shown below consists of arcs of four semi-circles with centers at the midpoints of the sides of a square. The square measures 20 cm by 20 cm. Find the area bounded by these circular arcs shaded in the figure shown. ## 03 Area enclosed by pairs of overlapping quarter circles

Example 03
The shaded regions in the figure below are areas bounded by two circular arcs. The arcs have center at the corners of the square and radii equal to the length of the sides. Calculate the area of the shaded region.

## 02 Area bounded by arcs of quarter circles

Example 02
Arcs of quarter circles are drawn inside the square. The center of each circle is at each corner of the square. If the radius of each arc is equal to 20 cm and the sides of the square are also 20 cm. Find the area common to the four circular quadrants. See figure below. ## 01 Arcs of quarter circles

Example 01
The figure shown below are circular arcs with center at each corner of the square and radius equal to the side of the square. It is desired to find the area enclosed by these arcs. Determine the area of the shaded region. 