Exercise 6. Triangles How many sides has a triangle? How many angles? Tri-angle means three angles. A triangle is a figure bounded by three straight lines or sides. Give examples of triangles which you have noticed in any of the objects in your community, such as the gable on a house. Area of a Triangle Draw a triangle of any shape. Then draw dotted lines parallel to two of the sides as shown in the illustration. What shaped figure is formed by the addition of the dotted lines? Cut the parallelogram along the middle line and compare the sizes of the two triangles. The first triangle is what part of the area of the parallelogram? The area of the parallelogram has been shown to be equal to the product of the base and the altitude. The area of a triangle is equal to one-half the product of its base and altitude.1 Exercise 7 1. Find the area of a triangle with a base of 8 inches and an altitude of 7 inches. Solution: The product of the base and altitude=56. of the product of the base and altitude=X56—28. Therefore the area of the triangle =28 square inches. 1Some pupil may also be able to show that the area of a triangle is equal to one-half of the base multiplied by the altitude or one-half of the altitude multiplied by the base. Find the areas of triangles having the following dimensions: 20 FT 8. The width of the gable of a barn is 20 feet and the altitude is 7 feet. Find the number of square feet in the area of the gable. Find the total area of the end of the barn, using the dimensions shown in the illustration. 9. A triangular plot of ground has a base of 20 rods and an altitude of 16 rods. How many acres are there in this plot of ground? 10. A triangular field is 20 rods long and its greatest width is 8 rods. How many square rods does it contain? Express the area of this field in acres. EHS. 11. A pennant in the shape of a triangle is 24 inches long and 8 inches wide at one end. How many square inches of felt are there in the pennant? 12. The gable of a house is 28 feet wide and 14 feet high. How many square feet are there in the area of this gable? Draw a diagram to show the shape of the gable. 13. A triangular city lot is 166 feet long and 90 feet wide. How many square feet does it contain? What part of an acre (43,560 sq. ft.) is it? 14. The width of the gable of a barn is 30 feet and the altitude 12 feet. Find the number of square feet in the gable of this barn. Exercise 8. Volumes of Rectangular Solids A solid is a figure having three dimensions, length, breadth and depth (or thickness). A rectangular solid is a solid with six rectangular faces. 1. Find the volume of a rectangular solid 4 feet long, 3 feet wide and 2 feet high. 2. How many square feet are there in the base of this solid? 3. How many cubic feet are there in a layer one foot deep? 4. How many such layers are there in the solid? 5. How many cubic feet are there in the volume of the rectangular solid? The volume of a rectangular solid is equal to the number of cubic units in a layer one unit high, multiplied by the number of layers. This principle is often expressed in the shorter form: The volume of a rectangular solid is equal to the product of the three dimensions. 6. Find the number of cubic feet in a room 12 feet long, 9 feet wide and 9 feet high. 12 cubic feet of air weigh a pound. How much will the air in such a room weigh? 7. An iceman brought me a rectangular piece of ice 11 inches long, 11 inches wide and 10 inches high for 50 pounds. Find the fraction of a cubic foot in this piece of ice. If ice weighs 57.5 pounds per cubic foot, how much under weight was the piece? 8. A farmer measured a rectangular crib of corn on the ear. The length was 16 feet 4 inches, the width 10 feet 7 inches and the depth of the corn 8 feet 2 inches. Find the number of bushels of corn in this crib, counting 4000 cu. in. per bushel. 9. Find the number of bushels of wheat (2150.42 cubic inches) in a bin 11 feet 5 inches long, 8 feet 6 inches wide and 7 feet 8 inches deep? 10. Find the number of bushels of wheat which can be loaded into a freight car 37 feet 6 inches long, 8 feet 6 inches wide and 5 feet in depth? 11. How many tons of coal can be loaded into a flat car 40 feet long, 8 feet 6 inches wide and 4 feet 6 inches in depth, allowing 38 cubic feet per ton? 12. Find the cost of excavating a basement 22 feet 6 inches long, 18 feet wide and 4 feet 6 inches deep at 50 cents per cubic yard. 13. Find the number of gallons (231 cubic inches) in a rectangular tank 6 feet long, 2 feet 2 inches wide and 2 feet 4 inches deep. 14. A hay mow 30 feet long and 16 feet 6 inches wide is filled with hay to an average depth of 10 feet. How many tons of hay are there in this mow, allowing 512 cubic feet per ton? Exercise 9. Bin Problems 1. Find the number of bushels of wheat in a bin 15 feet X 61 feet, filled to a depth of 5 feet. See page 236 for short method. 2. How many tons of hard coal can be put in a bin 10 feet long, 5 feet wide and 4 feet deep, allowing 35 cubic feet of space per ton? 3. How many bushels of potatoes will a freight car hold that is 37 feet long, 8 feet wide, if filled to a depth of 6 feet? A bushel of potatoes must be heaped to give full weight. A heaped bushel is equal to 2747.07 cubic inches. 4. A farmer built a potato bin 5 feet 6 inches long, 3 feet wide and 2 feet deep. How many bushels of potatoes did it hold when filled level full? 5. How many bushels of shelled corn can be put into a bin 6 feet 9 inches long, 3 feet 6 inches wide and 4 feet deep? (Express in fractions and use method on page 236.) 6. Harry's father built a coal bin 12 feet long, 7 feet wide and 5 feet high. At 36 cubic feet to the ton (for anthracite or hard coal), how many tons did this bin hold when filled? 7. Harry's uncle built a coal bin 14 feet X 8 feet X 5 feet. How much anthracite coal did it hold? 8. Mr. Field bought 80 tons of bituminous (soft) coal for use at his factory. His coal room was 28 feet long and 17 feet wide. What was the average depth of the coal in this room, allowing 42 cubic feet per ton? 9. Plan a bin to hold 10 tons of anthracite coal. Give length, width and depth. Prove that your bin will hold that amount. 10. Plan a bin to hold 20 bushels of potatoes. See Problem 3 for note on number of cubic inches in a bushel. 11. A bushel of corn in the ear occupies 4000 cubic inches of space. Find the number of bushels in a corn crib 16 feet 5 inches long, 8 feet 3 inches wide and filled to a depth of 10 feet 6 inches. 12. How many bushels of shelled corn can be put into a bin 10 feet 2 inches long, 7 feet 3 inches wide and 6 feet 10 inches deep? 13. Measure some bin and make a problem for class use based upon the actual measurements. 14. Plan a bin to hold 50 bushels of wheat |