Material such as paper, tin, cloth, etc., often comes in large sheets or pieces out of which smaller sheets or pieces of certain sizes are to be cut. We have paper cutting for our work; let us see if we are economical. While it is always possible to cut a sheet having an area of 48 sq. in. into two sheets each having an area of 24 sq. in., it is not possible to obtain two sheets having dimensions of 6" by 4" unless the original sheet has dimensions of 12" by 4" or dimensions of 6′′ by 8′′; therefore, great savings are made by using material of the proper size. To tell quickly how many rectangular sheets of a certain size can be cut from a rectangular sheet of another size, and also to determine how much waste 10" 10" there will be, we divide one of the dimensions of the large sheet by one of the dimensions of the small sheet to see how many it will make, then we divide the other dimension of the large sheet by the other dimension of the small sheet (in each case using as the divisor, that dimension which will leave the smallest remainder), and then we find the product of these two quotients. Sometimes it is most economical to cut the material so that there will be a strip remaining which will be wide enough to be used in the opposite direction. Exercise 45-Oral. 1. What is the area of a piece of tin 3′′ × 8"? The How many 11" 11" 4. Can we cut No. 1 into two pieces of the dimen 5. What dimensions should No. 1 have in order that we might cut it into two of No. 2? 6. What other dimensions might No. 1 have which would also enable us to cut it into two of No. 2? 7. Explain how you can find out how many 5′′ × 8′′ cards can be cut from a sheet 24" X 25" in size? 8. How can you tell if there will be any waste? 9. In each of the divisions, how can you tell which dimension of the small sheet should be used as the divisor? 10. How can we sometimes cut the material to better advantage than at first appears possible? Exercise 46 Written. 1. How many cards 4" X 6" can be cut from a sheet 18" X 20"? How much waste will there be? Show this by a diagram drawn to the scale of 1. 2. How many sheets of tin 8" X 10" can be cut from a sheet 18" X 20"? What are the dimensions of the remainder if there is one? 3. How many pieces of clcth 4" square can be cut from a piece 12" X 14"? What is the area of the remainder if there is one? 4. How many letter heads 81" X 11" can be cut from a sheet of paper 22" × 34"? What is the area of the remainder if there is one? 5. How many note heads 51" 81" can be cut from How many can be a sheet of paper 34" x 44"? How 6. How many circulars 6" X 91" can be printed at one time on a sheet of paper 24′′ × 38"? many square inches would be wasted? How many reams of 24" X 38" paper would be needed for 8,000 circulars? 7. The front and the back covers of a magazine are printed on one sheet of paper; if the finished magazine is 101" X 14", what is the size of the complete cover as it comes off the press? How many complete covers can be printed from a sheet 42" X 63"? 8. If both sides of a sheet of paper 24" X 38′′ are printed and made into a book 6′′ × 91′′ in size, how many pages will the book have? 9. How many letter heads 81" X 11" can be cut from a sheet 301⁄2′′ × 34′′? 10. A pan of candy is 20" x 30"; into how many pieces 3" X 11" can it be cut? LESSON 22 Finding the Volume of a Cylinder A "cylinder," as you know, is a solid bounded by a uniformly curved side and two parallel circular ends or bases of equal size. As in the case of the prism, we find the volume of a cylinder by finding how many cubic units there are in each layer of the cylinder, and multiplying this by the number of layers. Therefore, we find the area of one |