maxima and minima

55 - Greatest angle subtended by a picture

InverseTrigo-55 A man viewing a picture in the wall

Problem 55

The lower edge of the picture is a ft, the upper edge is b ft, above the eye of an observer. At what horizontal distance should he stand, if the vertical angle subtended by the picture is to be greatest?

 

26-27 Horizontal rod entering into a room from a perpendicular corridor

MaxMin 26 Horizontal rod entering from a corridor into a large room

Problem 26

A corridor 4 ft wide opens into a room 100 ft long and 32 ft wide, at the middle of one side. Find the length of the longest thin rod that can be carried horizontally into the room.

 

24-25 Largest rectangle inscribed in a circular quadrant

MaxMin 24 Rectangle inscribed in a circular quadrant

Problem 24

Rectangle inscribed in a circular quadrantFind the area of the largest rectangle that can be cut from a circular quadrant as in Fig. 76.

 

23 - Sphere cut into a pyramid

DiffCalc 020 Square pyramid inscribed in a sphere

Problem 23

A sphere is cut in the form of a right pyramid with a square base. How much of the material can be saved?

 

22 - Smallest cone that may circumscribe a sphere

MaxMin 22 Sphere inside a conical vessel

Problem 22

A sphere of radius a is dropped into a conical vessel full of water. Find the altitude of the smallest cone that will permit the sphere to be entirely submerged.

 

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

MaxMin 20 Horizontal pole passing through 90 degree corridors

Problem 20

A pole 27 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?

 

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

MaxMin 19 Route plan of a man on an island to a point on the straight shore

Route plan of a man on an island to a point on the straight shore

Problem 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

MaxMin 18 - A motorboat 20 miles away from a straight shore line

Problem 17

17-given-figure.jpgA 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

MaxMin 16 Light placed above the circle

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

 

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

MaxMin 14 Ladder from outside the wall to the house

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.

 

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