# 21 - 24 Solved problems in maxima and minima

**Problem 21**

Find the rectangle of maximum perimeter inscribed in a given circle.

# 18 - 20 Rectangular beam in maxima and minima problems

**Problem 18**

The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size.

# 15 - 17 Box open at the top in maxima and minima

**Problem 15**

A box is to be made of a piece of cardboard 9 inches square by cutting equal squares out of the corners and turning up the sides. Find the volume of the largest box that can be made in this way.

# 12 - 14 Rectangular Lot Problems in Maxima and Minima

**Problem 12**

A rectangular field of fixed area is to be enclosed and divided into three lots by parallels to one of the sides. What should be the relative dimensions of the field to make the amount of fencing minimum?

# 09 - 11 Rectangular Lot Problems in Maxima and Minima

**Problem 9**

What should be the shape of a rectangular field of a given area, if it is to be enclosed by the least amount of fencing?

# Area of Regular Six-Pointed Star

**Problem**

Find the area of the regular six-pointed star inscribed in a circle of radius 20 cm.

# Circle Tangent Internally to Another Circle

# Area of Regular Five-Pointed Star

**Problem**

Find the area of the regular five-pointed star inscribed in a circle of radius 20 cm.

# Relationship Between Central Angle and Inscribed Angle

Central angle = Angle subtended by an arc of the circle from the center of the circle.

Inscribed angle = Angle subtended by an arc of the circle from any point on the circumference of the circle. Also called *circumferential angle* and *peripheral angle*.

Figure below shows a central angle and inscribed angle intercepting the same arc AB. The relationship between the two is given by

if and only if both angles intercepted the same arc. In the figure below, θ and α intercepted the same arc AB.