3. Statement of rules as formulas. We have seen that it is often very convenient to write rules as formulas. The formula is shorter, more easily remembered, and easier to use in calculating. It is important to know (a) How to translate a rule or other statement in words into a formula. (b) How to translate a formula into an English sentence. (c) How to use a formula in calculating. We shall now have practice in dealing with formulas in the first of these ways, and later in each of the other two ways. EXAMPLE. If a body is let fall from the top of a tower the number of feet, s, that it falls in a certain number of seconds, t, is 16.1 times the square of the number of seconds. This principle is briefly stated in the formula s=16.1 t2. Exercise 2 Write the formulas which give the answers to the following exercises: 1. Find the selling price, s, when the cost is c and the gain is g. 2. Find the average, m, of six numbers a, b, c, d, e, and f. 3. Using the same letters that you used in percentage problems, find (a) the percentage, given the rate and the base; (b) the base, given the rate and the percentage; (c) the rate, given the base and the percentage. 4. Write a formula for finding (a) The quotient, q, given the dividend, D, and the divisor, d. (b) The dividend, D, given the divisor, d, the quotient, q, and the remainder, r. (c) One factor, F, when the other factor, f, and the product, p, are given. 5. Write a formula for finding (a) The number of feet, f, of bookshelves in a room which contains a shelves each n feet long, and 4 shelves each 6 feet long. (b) The number of seats, s, in a room which contains x double desks (desks that will seat 2 persons), a single desks, and 1 chair. (c) The number of bushels of corn, n, that a farmer raises, if he gets 40 bushels an acre from x acres, and 60 bushels an acre from y acres. Write formulas for the answers to the following exercises: 6. A tank is being filled by two pipes. It now contains 400 gallons. How many gallons will it contain after 45 minutes if one pipe brings in a gallons a minute, and the other b gallons a minute? 7. In a certain year the farmers of Iowa raised a acres of corn which gave an average yield of b bushels per acre. The next year they raised A acres and the yield increased to B bushels per acre. The total increase in the yield was I bushels. Find I. 8. There were 200 pounds of flour in a barrel. There have been used from it p pounds a day for 10 days, pounds a day for d days. How much remains? represent the number of pounds that remain. and 5 Let b The 9. The weight of 40 eggs and a basket is x ounces. basket weighs y ounces. How many ounces do the eggs weigh? Find the weight, w, of one egg. 10. A board is n feet long. From it are cut 7 pieces of equal length, leaving a piece 2 feet long. What is the sum of the lengths of the 7 equal pieces? Find the length, l, of one of the equal pieces. 11. A boy now has $15.35. He makes $1.25 a day and spends $1.05. What is the sum of money, s, that he has at the end of d days? 12. A boy solves n problems on Tuesday and 20% more on Wednesday. Find the number, N, that he solves on Wednesday. 13. Let x be the total number that he solves on both Tuesday and Wednesday. Find x. 14. The cost of an article is c. The selling price is 10% less than the cost. Find the selling price. 15. A bank lends $3000 at n% interest. Find the interest, i, for 1 year; for x years. 16. The cost of an article is c dollars. It is marked to sell at 25% above the cost. The article becomes damaged and is sold for $2 less than the marked price. The selling price is s dollars. Find s. 17. A bookshelf will hold x books 1 inch thick, and y books 1 inches thick. The shelf is I feet long. Find l. 18. General admission to a baseball game is 50 cents, reserved seats 75 cents. There are x general admissions sold, and y reserved seats. Find the total receipts, r. 19. The rule for making coffee in camp is "One spoonful for each person and one for the pot." Write this rule as a formula where n is the number of spoonfuls and p is the number of persons. 20. A cistern contains 125 barrels of water. One pipe carries water into it at the rate of m gallons an hour, and another takes water out at the rate of n gallons an hour. After h hours there are g gallons of water in the cistern. Find g. 21. Find Mr. Blank's income, I, if he has invested a dollars at 5%, b dollars at 6%, and c dollars at 7%, and has a salary of d dollars. 22. A string whose length is l is 3 inches too short to go around a rectangle a"-b". Find l. 4. Translating formulas into English sentences. To understand and explain a formula it is often necessary to be able to translate it into an English sentence. EXAMPLE. We have had the formula, D=dq+r, for finding the dividend when the divisor, quotient, and remainder are given. This formula may be stated: The dividend equals the product of the divisor and quotient plus the remainder. Exercise 3 State in words the rule expressed by each of the following formulas : 1. Aab. The formula for the area of a parallelogram. ab 2. A 2 3. Aab+b). The formula for the area of a trapezoid. 4. A=πr2. 2 The formula for the area of a circle. 5. As2. The formula for the area of a square. This and the next two are percentage formulas. 6. prb. 9. 10. P P. i=prt. This and the next three are interest formulas. = 13. V lwt. The formula for finding the volume of a rectangular solid. 14. V=e3. The formula for finding the volume of a cube. 15. d-rt. The formula for finding the distance when the rate and time are given. 16. grc. Gain, rate, and cost formula. 17. 8=c+rc. The formula for finding the selling price, given the cost and the rate of gain. 5. Substitution in formulas. The engineer and mechanic make much use of the formula in calculation. In order to use the formula it is necessary to be able to substitute numbers for the letters and to simplify the results. EXAMPLE. A bomb is dropped from an airplane. How far will it fall in 20 seconds? The answer to this question may be found by using the formula given on page 4, s=16.12. Substituting in this formula we have s=16.1X202=6440, the number of feet the bomb falls in 20 seconds. Exercise 4 1. Use the formula for finding the volume, V, of a rectangular solid when the length, l, the width, w, and the thickness, t, are given, and find V when 1, w, and t have the following values. Copy and fill out this table. 2. The product 8X8X8 is written 83, which is read "8 cube." The product nXnXn is written n3, which is read "n cube." The formula Ve gives the volume of a cube whose edge is e. Find V for the following values of e: 6; 12; 25; .7; 8.6; 7.02; 7; 31. 3. Show that the area of Figure 1 is given by the formula A=a2-b2. Find A when a and b have the following values. Copy and fill out this table. |