# Maxima and Minima

## 16 - Light placed above the center of circular area

**Problem 16**

A light is to be placed above the center of a circular area of radius a. What height gives the best illumination on a circular walk surrounding the area? (When light from a point source strikes a surface obliquely, the intensity of illumination is

where $\theta$ is the angle of incidence and $d$ the distance from the source.)

## 14-15 Ladder reaching the house from the ground outside the wall

**Problem 14**

A wall 10 ft high is 8 ft from the house. Find the length of the shortest ladder that will reach the house, when one end rests on the ground outside the wall.

## 13 - Sphere cut into a circular cone

## 12 - Cone of maximum convex area inscribed in a sphere

**Problem 12**

Find the altitude of the circular cone of maximum convex surface inscribed a sphere of radius a.

## 11 - Triangular gutter of maximum carrying capacity

**Problem 11**

A gutter having a triangular cross-section is to be made by bending a strip of tin in the middle. Find the angle between the sides when the carrying capacity is to a maximum.

## 10 - Largest conical tent of given slant height

**Problem 10**

Find the largest conical tent that can be constructed having a given slant height.

## 06-09 Trapezoidal gutter of greatest capacity

**Problem 06**

A trapezoidal gutter is to be made, from a strip of metal 22 inches wide by bending up the edges. If the base is 14 inches wide, what width across the top gives the greatest carrying capacity.

## 04-05 Stiffness and strength of timber beam

**Problem 4**

The stiffness of a rectangular beam is proportional to the breadth and the cube of the depth. Find the shape of the stiffest beam that can be cut from a log of a given size.

## 03 - Heaviest cylinder that can be made from a shot

**Problem 3**

Find the weight of the heaviest circular cylinder can be cut from a 16-lb shot.

## 02 - Cylinder of maximum convex area inscribed in a sphere

**Problem 02**

A cylinder is inscribed in a given sphere. Find the shape of the cylinder if its convex surface area is a maximum.