maxima and minima

72 - 74 Light intensity of illumination and theory of attraction

DiffCalc 022 Light illuminating a circular area

Problem 72

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 θ is the angle of incidence and d the distance from the source.)

 

Solution:

69 - 71 Shortest and most economical path of motorboat

DiffCalc 021 Trip diagram for minimum cost

Problem 69

A man on an island 12 miles south of a straight beach wishes to reach a point on shore 20 miles east. If a motorboat, making 20 miles per hour, can be hired at the rate of $2.00 per hour for the time it is actually used, and the cost of land transportation is $0.06 per mile, how much must he pay for the trip?

 

66 - 68 Maxima and minima: Pyramid inscribed in a sphere and Indian tepee

DiffCalc 020 Square pyramid inscribed in a sphere

Problem 66

Find the largest right pyramid with a square base that can be inscribed in a sphere of radius a.

 

64 - 65 Maxima and minima: cone inscribed in a sphere and cone circumscribed about a sphere

Problem 64

A sphere is cut to the shape of a circular cone. How much of the material can be saved? (See Problem 63)

 

62 - 63 Maxima and minima: cylinder inscribed in a cone and cone inscribed in a sphere

DiffCalc 019 Cylinder inscribed in a cone

Problem 62

Inscribe a circular cylinder of maximum convex surface area in a given circular cone.

 

60 - 61 Maxima and minima problems of a folded page

Problem 60

Figure 41One corner of a leaf of width a is folded over so as just to reach the opposite side of the page. Find the width of the part folded over when the length of the crease is a minimum. See Figure 41.

 

58 - 59 Maxima and minima: cylinder surmounted by hemisphere and cylinder surmounted by cone

Problem 58

For the silo of Problem 57, find the most economical proportions, if the floor is twice as expensive as the walls, per unit area, and the roof is three times as expensive as the walls, per unit area.

 

56 - 57 Maxima and minima problems of square box and silo

DiffCalc 017 Square box made from different materials

Problem 56

The base of a covered box is a square. The bottom and back are made of pine, the remainder of oak. If oak is m times as expensive as pine, find the most economical proportion.

 

53 - 55 Solved Problems in Maxima and Minima

DiffCalc 016 Largest rectangle inscribed in a circular quadrant

Problem 53
Cut the largest possible rectangle from a circular quadrant, as shown in Fig. 40.

 

50 - 52 Nearest distance from a given point to a given curve

DiffCalc 015 Nearest distance from the ellipse to a given point

Problem 50

Find the shortest distance from the point (4, 2) to the ellipse x2 + 3y2 = 12.

 

Syndicate content

 

 

Featured Free Magazine

Featured Free Download