Maxima and Minima

Length of one side for maximum area of trapezoid (solution by Calculus)

BC of trapezoid ABCD is tangent at any point on circular arc DE whose center is O. Find the length of BC so that the area of ABCD is maximum.

55 - Greatest angle subtended by a picture

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

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

Problem 24
Find the area of the largest rectangle that can be cut from a circular quadrant as in Fig. 76.

23 - Sphere cut into a pyramid

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

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

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?


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