9. Flow from a Tank. A tank 28 feet 9 inches high, has a discharge pipe on the side at a distance of 3 inches from the bottom. Compute the rate of flow from the discharge pipe when the surface of the water is 3 feet below the top of the tank. The head in this instance is the distance of the discharge pipe from the surface of the water. 10. Flow Computed from Pressure Gage. When the pressure of the water at the place of discharge is denoted by a gage, the rate of discharge may be determined by the formula V=12.16VP in which V = velocity of flow in feet per second; P=pressure in pounds per square inch, as read from the gage. Write the law. 11. By the formula of Problem 10, compute the flow in feet per second, under a pressure of 45 lbs. per square inch. 12. If the pressure is increased to 225 lbs. per square inch, by high-pressure pumps, how much is the flow increased over that in Problem 11? 13. Diameter of a Chimney. The diameter of a chimney in inches equals 4, plus 13.54 times the square root of the effective area in square feet. Write the formula, denoting diameter by D and effective area by E. 14. Compute the diameter of chimney required for an effective area of 6.57 square feet. 15. Effective Area of a Chimney. The effective area of a chimney may be determined by the formula, Copy the formula and write the law. 16. By the formulas in the two preceding problems, compute and fill in, the omitted entries in the following table: * Entries in this column are from Swingle's 20th Century Handbook. 17. Volume of an Iron Ball. The volume or solid contents of an iron ball in cubic inches equals .5236 times the cube of the diameter of the ball in inches. Formulate the law. 18. By the formula of Problem 17, compute the volume in cubic inches, of an iron ball 11 inches in diameter. 19. Weight of a Cast-iron Ball. If the iron ball of Problem 18 weighs .26 lb. per cubic inch, formulate and compute its weight to the nearest pound. 20. Time of Falling. The time in seconds, in which a body will fall under gravity from any height, equals the square root of the quotient of the height in feet divided by 16.08. Formulate the law and compute the number of seconds required by a body to fall to the ground from a height of 1000 feet. 21. Time of Beat of a Pendulum. The time in seconds, in which a pendulum will make one beat, equals 3.1416 times the square root of the quotient of the length of the pendulum in feet divided by 32.16. Write the formula. 22. Compute the time required for a pendulum 39.102 inches long, to make one beat. 23. Board Feet in a Log. The number of feet of boards which can be cut from a 12-foot log equals the number of square inches in half the square of its diameter. How many feet of boards can be cut from a 12-foot log, 18 inches in diameter? 24. Altitude of an Equilateral Triangle. An equilateral triangle is one whose three sides are the same length. The alti tude is the perpendicular distance from any vertex to the side opposite. The altitude of an equilateral triangle equals the length of a side of the triangle, times the square root of 3. Draw a figure in which the side is denoted by S and the altitude by h. Write the formula. 25. Compute the altitude of an equilateral triangle whose side measures 125 inches. 26. Area of a Triangle. If the sides of a triangle are denoted by a, b, and c, and half the sum of the sides by s, the area is expressed by the formula A =√s(s—a) (s—b) (s—c) Draw the triangle to scale and compute the area when the sides are 18, 21, 26. CHAPTER III FRACTIONS SECTION 1. REDUCTION. SECTION 2. ADDITION AND SUBTRACTION. SECTION 3. MULTIPLICATION. SECTION 4. DIVISION. SECTION 5. APPLIED PROBLEMS. 49. Illustration and Definition. A foot rule is marked off into twelve equal parts called inches. Not only is the foot marked off into equal parts, but each inch is divided into 2 equal parts, 4 equal parts, 8 equal parts, 16 equal parts, etc. Each of the 2 equal parts is called a half inch; each of the 4 equal parts is called a quarter inch, and so on. The preceding numbers,, fa, fa, fa, 1, 1, and are called fractions, because they denote a part or fraction of some thing which has been divided. We therefore have the following definition. A fraction is a number which denotes one or more of the equal parts into which some thing has been divided. You will observe that a fraction consists of three things: (1) The fraction line, which may be horizontal or oblique, as or 3/4. (2) The numerator or number above the line, which denotes how many of the equal parts are taken. (3) The denominator or number below the line, which denotes into how many equal parts the unit has been divided. Thus in, the denominator 12 denotes a division into 12 equal parts, and the numerator 5 denotes that 5 of these parts are taken. A fraction therefore is an indicated division of the numerator by the denominator. While for convenience, the oblique fraction line is sometimes used, your work will be more easily read if you use the horizontal instead of the oblique line. § 1. REDUCTION 50. Reduction to Lowest Terms. A fraction is said to be in its lowest terms when no number will evenly divide both numerator and denominator. Thus is in its lowest terms because no number will evenly divide both its numerator and denominator. On the contrary, is not in its lowest terms because 3 will evenly divide both numerator and denominator. Thus, both terms being divided by 3. We therefore have the following RULE FOR REDUCING TO LOWEST TERMS. To reduce a fraction to its lowest terms, divide both its numerator and denominator by any number that is evenly contained. Continue this process until further division is impossible. Thus 196 49 7 The first divisor was 4, the second 7. Further division is impossible. |