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

$I = \dfrac{k\sin \theta}{d^2}$

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

Problem 13
A sphere is cut in the shape of a circular cone. How much of the material can be saved?

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.