Trigonometric Functions

20-21 Width of the second corridor for a pole to pass horizontally

Problem 20
A pole 24 feet long is carried horizontally along a corridor 8 feet wide and into a second corridor at right angles to the first. How wide must the second corridor be?

Problem 21
Solve Problem 20 if the pole is of length $L$ and the first corridor is of width $C$.

19 Direction of the man to reach his destination as soon as possible

Route plan of a man on an island to a point on the straight shoreProblem 19
A man on an island a miles south of a straight beach wishes to reach a point on shore b miles east of his present position. If he can row r miles per hour and walk w miles per hour, in what direction should he row, to reach his destination as soon as possible? See Fig. 57.


17-18 A man in a motorboat needs to catch a bus

17-given-figure.jpgProblem 17
A man in a motorboat at A receives a message at noon, calling him to B. A bus making 40 miles per hour leaves C, bound for B, at 1:00 PM. If AC = 30 miles, what must be the speed of the boat, to enable the man to catch the bus?

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.