286 37. Optimal garden A rectangular flower garden with an area of 30 m is surrounded by a grass border 1 m wide on two sides and 2 m wide on the other two sides (see figure). What dimensions of the garden minimize the combined area of the garden and borders? 3.5 mi Power station Shore 8 mi 29. Laying cable again Solve the problem in Exercise 28, but this time minimize the cost with respect to the smaller angle 0 between the underwater cable and the shore. (You should get the same answer.) 2 m Flower garden 30. Shortest ladder A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal 38. Rectangles beneath a line ground have infinite extent. 31. Shortest ladder-more realistic An 8-ft-tall fence runs parallel to the wall of a house at a distance of 5 ft. Find the length of the a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the ori- gin on the line y = 10 – 2x. What dimensions maximize the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house is 20 ft high and the horizontal ground extends 20 ft from the fence. area of the rectangle? What is the maximum area? b. Is it possible to construct a rectangle with a greater area than that found in part (a) by placing one side of the rectangle on the line y = 10 the positive x- and y-axes? Find the dimensions of the rect- angle of maximum area that can be constructed in this way. 2x and the two vertices not on that line on 32. Circle and square A piece of wire of length 60 is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to (a) maximize and (b) minimize the combined area of the circle and the square? 9. Designing a box Two squares of length x are cut out of adjacent corners of a 18" × 18" piece of cardboard and two rectangles of length 9 and width x are cut of out the other two corners of the cardboard (see figure). The resulting piece of cardboard is then folded along the dashed lines to form an enclosed box (see figure). Find the dimensions and volume of the largest box that can be formed in this way. 33. Maximum-volume cone A cone is constructed by cutting a sector from a circular sheet of metal with radius 20. The cut sheet is then folded up and welded (see figure). Find the radius and height of the cone with maximum volume that can be formed in this way. 20 20 34. Slant height and cones Among all right circular cones with a slant height of 3, what are the dimensions (radius and height) that maximize the volume of the cone? The slant height of a cone is the distance from the outer edge of the base to the vertex. 40. Folded boxes a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid, Find the volume of the largest box that can be formed in this way. 35. Optimal soda can a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm that minimize the surface area. b. Real problem Compare your answer in part (a) to a real soda can, which has a volume of 354 cm, a radius of 3.1 cm, and a height of 12.0 cm, to conclude that real soda cans do not seem b. Squares with sides of length x are cut out of each corner of a square piece of cardboard with sides of length l. Find the vol- ume of the largest open box that can be formed in this way. ст,