Problem 64 A sphere is cut to the shape of a circular cone. How much of the material can be saved? (See Problem 63).
Problem 62 Inscribe a circular cylinder of maximum convex surface area in a given circular cone.
Problem 60 One 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.
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.
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.
Problem 53 Cut the largest possible rectangle from a circular quadrant, as shown in Fig. 40.
Problem 50 Find the shortest distance from the point (4, 2) to the ellipse x2 + 3y2 = 12.
Problem 48 Find the shortest distance from the point (5, 0) to the curve 2y2 = x3.
Problem 46 Given point on the conjugate axis of an equilateral hyperbola, find the shortest distance to the curve.
Problem 43 A ship lies 6 miles from shore, and opposite a point 10 miles farther along the shore another ship lies 18 miles offshore. A boat from the first ship is to land a passenger and then proceed to the other ship. What is the least distance the boat can travel?
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