literal number p then stands for the corresponding numbers 30, 60, 90, etc. A number for which a literal number stands is a value of the literal number. 32. The meaning of the word formula. The statement p=30 Xn expresses the equality of the two numbers p and 30 Xn, and is therefore an equation. Exercise 2, k (830), shows that the equation p=30 Xn enables us to determine the price of any given number of dozens of oranges selling at 30 cents a dozen. When an equation is used to state briefly a rule for obtaining numerical facts from other related facts it is a formula. Thus, p=30Xn is a formula. The following exercises teach how to make formulas": EXERCISES 1. a. Let n represent the number of articles purchased, let p be the price of each, in cents, and let c be the total cost in cents. In the table to the right find the values of c for given values of n and p. Thus, when n=1, p=10, c=1X10=10, etc. b. Make a formula for finding the cost c for n articles each sold at p cents. c. Show that all the facts found in Exercise 1, a, can be obtained from the formula c=nXp. 8 16 4 4 5 18 2. a. Make a table giving the number of inches, i, corresponding to the number of feet, f, letting s take the values 1, 2, 3, .... b. Make a statement expressing in words, the number of inches in terms of the number of feet. Translating this statement into symbols, make a formula for finding the values of i, Exercise 2, a, corresponding to values of f. c. Using the facts given in the table of Exercise 2, a, make a graph of the formula i=12Xf. 3. a. A cubic foot of water weighs 62.5 pounds. Show that the number of pounds (weight) w, of a volume of v cubic feet is given by the formula w=62.5 Xv. Translate this formula into words. b. Tabulate corresponding values of v and w, Exercise (a) and make a graph of the equation w=62.5 Xv. 4. a. A train traveling over equal distances in equal timespaces is said to have uniform motion. The distance traveled in the unit of time is the velocity (speed, or rate) of the train. Complete the table at the right, stating values of the distances, d, corresponding to values of the time, t, for a train having a velocity, r, of 20 t d miles an hour. 1 20 1 X 20=20 2 20 20 20 t 20 1 20 r 15 b. In the table below find the values of d corresponding to given values of r and t. C. Write a formula from which to find the distance, d, of a train traveling t d at a rate, r, for t hours. 1 30 1x30=30 5. Make a formula for finding the 6 number of feet, f, from a number of 12 40 yards, y. Translate this formula into 8 20 words. 15 10 t 16 6. A boy earns d dollars a week for 18 n weeks. State a formula for finding t his total earnings t. 7. If n oranges cost C cents, state a formula for finding the cost, c, of one orange. r 8. Make a formula for finding the number of gallons n, of oil, which pass through a pipe in m minutes, at the rate of 2 gallons a minute. 9. A river flows at the rate of r mi. an hour. Find the distance, d, an object will float in t hours. 10. Make a formula for finding the quotient, q, when the dividend is D and the divisor d. 11. A boy is n years old. What was his age, a, 5 years ago? What will be his age, A, 5 years from now? 12. A girl saves $2 a week for n weeks. Make a formula for finding her total savings x. 13. A girl receives 10 cents each time she does the shopping for her mother. Make a formula for finding the amount, t, which she earns in n days, if she goes shopping twice a day. 33. Various uses of the formula. As we go on in the study of mathematics we shall meet formulas frequently because the formula saves time and effort in the solution of many problems. Formulas are used in other school subjects. Thus, in science the formula helps us to find the distance a particle falls in a given time, and in physics many laws are stated as formulas. The formula plays an important part in every-day life. The automobile owner determines the horsepower of his gasoline engine by means of a formula, the business man may use it to compute interest on a sum of money, and the machinist to find the length of belting connecting two pulleys. Scientists, engineers, machinists, surveyors, insurance men, and many others use formulas, and should know the correct way in which to work with them. The importance of the formula is one of many reasons for studying mathematics. PERIMETER FORMULAS 34. Polygons. On notebook paper mark points A, B, C, D, E, and F, as shown in the E diagram (Fig. 59) and join them with line segments. The figure thus formed is called a polygon. The word polygon comes from the Greek and C means a figure with many angles. B What is the least number of segFig. 59 ments needed to form a polygon? The points A, B, C, etc., are the vertices, the segments AB, BC, CD, etc., are the sides of the polygon. Measure the sides and find the sum AB+BC+CD+DE+EF+FA. The sum of the sides of a polygon is the distance around, or the perimeter of the polygon. A polygon is called a triangle, quadrilateral, pentagon, hexagon, according as it has 3, 4, 5, 6, .... sides. A polygon is equilateral when all the sides are of equal length. . EXERCISES 1. Measure each side of triangle ABC (Fig. 60) and find the perimeter by adding the lengths. с Denoting the perimeter by p, state the result in the form of an equation. When the sides are not measured, the B perimeter, p, of triangle ABC is given by Fig. 60 the formula p=a+b+c. a A с 2. Without measuring the sides find the perimeter of the equilateral triangle (Fig. 61) and state the result in the form of a formula. а a 3. What is the perimeter, p, of a triangle whose sides are 3x ft., 4x ft., and 5x feet? State the result in the form of an equation. Fig. 61 4. For each of the following equations sketch at least one polygon whose perimeter is expressed by the equation: 1. p=3x. 2. p=5x. 3. p=6x. 4. p=8x. Suggestion: Change 3.c to x+x+x, 6x to 3x+2x+x or 4x+x+x, etc. 5. Find the perimeter of each of the polygons in Exercise 4 when x=2.75 cm. 6. Write the equations in Exercise 4 when the perimeter p=148. 7. The perimeter of an equilateral hexagon is 120 inches. Find the length of a side. Solution: Let x be the number of inches contained in the side. Show that 120=6x. 8. The perimeter of an equilateral quadrilateral is 24. Find a side by means of an equation, as shown in Exercise 7. Find 9. The perimeter of an equilateral pentagon is 60. each side. 10. The perimeter of an equilateral decagon (10-sided polygon) is 28. Find each side. 35. Symbol for hence and therefore. The symbol used to denote hence or therefore is .. |