03 - Heaviest cylinder that can be made from a shot Jhun Vert Fri, 05/08/2020 - 03:01 pm

Problem 3
Find the weight of the heaviest circular cylinder can be cut from a 16-lb shot.

Cylinder of maximum volume and maximum lateral area inscribed in a cone

A right circular cylinder of radius r and height h is inscribed in a right circular cone of radius 6 m and height 12 m.

Part 1: Determine the radius of the cylinder such that its volume is a maximum.
A. 2 m
B. 4 m
C. 3 m
D. 5 m

Part 2: Determine the maximum volume of the cylinder.
A. 145.72 m3
B. 321.12 m3
C. 225.31 m3
D. 201.06 m3

Part 3: Determine the height of the cylinder such that its lateral area is a maximum.
A. 10 m
B. 8 m
C. 6 m
D. 4 m

62 - 63 Maxima and minima: cylinder inscribed in a cone and cone inscribed in a sphere Jhun Vert Wed, 05/06/2020 - 08:48 pm

Problem 62
Inscribe a circular cylinder of maximum convex surface area in a given circular cone.

Problem 63
Find the circular cone of maximum volume inscribed in a sphere of radius a.

53 - 55 Solved Problems in Maxima and Minima

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

Problem 54
A cylindrical tin boiler, open at the top, has a copper bottom. If sheet copper is m times as expensive as tin, per unit area, find the most economical proportions.

Problem 55
Solve Problem 54 above if the boiler is to have a tin cover. Deduce the answer directly from the solution of Problem 54.

38 - 40 Solved problems in maxima and minima

Problem 38
A cylindrical glass jar has a plastic top. If the plastic is half as expensive as glass, per unit area, find the most economical proportion of the jar.

Problem 39
039-figure-39.jpgA trapezoidal gutter is to be made from a strip of tin by bending up the edges. If the cross-section has the form shown in Fig. 38, what width across the top gives maximum carrying capacity?

Problem 40
Solve Ex. 39, if the strip is 11 inches wide and the base is 7 inches wide.

29 - 31 Solved problems in maxima and minima

Problem 29
The sum of the length and girth of a container of square cross section is a inches. Find the maximum volume.

Problem 30
Find the proportion of the circular cylinder of largest volume that can be inscribed in a given sphere.

Problem 31
In Problem 30 above, find the shape of the circular cylinder if its convex surface area is to be a maximum.

13 Review Problem - Volume of water inside the Venturi meter

Problem 13
The accompanying figure represents the longitudinal view of a Venturi meter, a device designed to measure the flow of water in pipes. If the throat of the of the meter is 6 in. long and has an inside diameter of 4 in., find the volume of water in the meter which is used in 12-in. pipe line if the altitudes of the tapering parts are in the ratio 1:3 and the smaller altitude measures 12 in.