please help me out with a paper explanation please, I will really appreciate it. I am having a hard time with some of the questions

14. A piece of plexiglass is in the shape of semicircle with radius 2 m. Determine the dimensions of the rectangle with the greatest area that can be cut from the piece of plexiglass. (ANS: 2v2 m by v2 m) Part D: Surface Area/Volume 15. A box with a square base and open top must have a volume of 4000 cm. Find the dimensions of the box that minimizes the amount of material used. (ANS: 20 cm by 20 cm by 10 cm) 16. A box with an open top is to be constructed from a square piece of cardboard, 3 m wide, by cutting out a square from each of the four corners and bending up the sides, as indicated in the figure. Find the largest volume that such a box can have. (ANS: 2 m?) 17. A can is to be made to hold a litre of oil. Find the radius of the can that will minimize the cost of the metal to make the can. (1 L = 1000 cm3) (ANS: (300 = 5.4 cm) Part c 18. A rectangular backyard playpen for a child to be enclosed with 16 m of flexible fencing, What dimensions of the rectangle will provide the maximum area for the child to play? (ANS: 4 m by 4 m) 19. A rectangular corral is to be enclosed along the side of a horse barn with the barn serving as one side of the corral. What dimensions of the corral, using 40 m of fencing, will enclose the maximum area for the horses? (ANS: 20 m by 10 m) 20. A rectangular garden plot requires an area of 32 m for the variety of vegetables that are to be planted, What dimensions of the plot will use the least amount of fencing to enclose the garden? (ANS: 4v2 m x 4v2 m) 21. a) A county fair has a holding area for the prize sheep that are entered in a contest. The holding area is made up of 12 identical pens arranged in a three by four grid. If 100 m of fencing is available, what dimension of pen will maximize the total holding area? (ANS: ~ m x ) b) If the pens were arranged in a two by six grid, what dimensions would maximize the holding area? :am x 5 m) 22. A soda cracker package (in the shape of a rectangular prism) is to be constructed with a square base. The total capacity of the package must be 1000 cm*. What dimensions provide the minimum surface area? (ANS: 10 cm by 10 cm by 10 cm) 23. An open metal box for removing ashes from a fireplace is to be constructed from a rectangular piece of sheet metal that is 1 m by 1.5 m. Squares are to be cut from each corner of the sheet metal, the sides folded upward to form the box, and then the seams welded. What is the maximum capacity of a box that is constructed in this way? (ANS: V = 0. 132 m )