and a and d are the extremes. Name the means and 1 4 the extremes in the following proportions: 2 8 6 18 m 5 15' n Y X . 92. Fundamental property of proportion. Write a 3 6 proportion, as Find the product of the means; 7 14 the product of the extremes. How do these products compare? Form another proportion, and compare the products of the means with the product of the extremes. The preceding examples illustrate the principle that in a proportion the product of the means is equal to the product of the extremes. This principle suggests a convenient method of solving a proportion for the unknown number. This method is to be used in the exercises below. EXERCISES Solve the following problems by means of proportions: 1. A boat runs 30 mi. in 3 hours. How many miles will it run in 8 hours? Solution: The ratio of the time of 3 hours to the time of 8 hours 3 is 8 100 TL Denoting the required distance by x, the ratio of the corre Since the product of the extremes, 3x, is equal to the product of the means, 8X30, we have 3.x=30X8, 10 30X8 .. I= 8 or - 80. 2. If 12 acres of land yield 440 bushels of corn, at the same rate of yield how many acres would yield 200 bushels? Suggestion: As in Exercise 1, state the proportion and then solve the equation. 3. A farm valued at $11,400 is taxed for $76.38. At the same rate what would be the tax on a farm valued at $14,250? 4. A boy pays 90 cents for 2 doz. oranges. What is the price of 20 oranges? 5. Five bars of soap are sold for 35 cents. At the same rate, find the price of 8 bars. 6. If the simple interest on a sum of money for 6 years is $200, what will be the interest for 10 years? 7. The dimensions of a rectangle are 9 inches and 5 inches. Find the width of a similar rectangle 12 inches long. Suggestion: Make a sketch before writing the equation. 8. The shadows of a pole and a 5-foot rod are respectively 80 feet and 7 feet. Make a sketch and then find the height of the pole by means of an equation. 9. If a bushel of shelled corn weighs 56 lb., how many ounces does a pint weigh? 10. The food parts of beef are protein, fat, and water. In 5 lb. of sirloin steak there are 13 oz. protein, 14 oz. fat, and 42 oz. of water, the remainder being waste material. How many ounces of protein, fat, and water are there in 3 lb. of sirloin steak? 11. If yd. of lace cost 63 cents, how many yards can be purchased for $3.50? 12. If 3 yd. of ribbon cost $2.70, what is the price of 14 yards? 13. If an automobile runs 16 mi. on a gallon of gasoline, how much gasoline will be consumed on a 350-mile trip? 14. If a stenographer writes 475 words in 3 minutes, how long will it take her to write 2000 words? 15. If a $64.80 tax is paid on property assessed at $2575, what tax should be paid on property assessed at $6000? 16. If 230 lb. of milk produce 8.3 lb. of butter fat, how many pounds of milk will be required to produce 35 lb. of butter fat? 17. If 86 lb. of metal make 15 castings, how much metal will be required to make 10 similar castings? 18. If 18 yd. of silk cost $48.50, what will 35 yd. cost? THE RIGHT-TRIANGLE METHOD OF FINDING UNKNOWN DISTANCES 93. Advantages of this method. Several methods have been used to determine unknown distances by indirect measurement. Each method is an improvement over the preceding methods. As we continue the study of mathematics, we also continue to improve our methods. In the method explained below the number of measurements required to solve the problem is reduced to a minimum. This also reduces the number of errors. The result does not depend on the accuracy of a drawing. For this reason it is generally used in practical work, such as surveying, where accuracy is important. It is called the right-triangle method. The method is made clear in $894–96. 94. Similar right triangles. On squared paper draw a right triangle (Fig. 177) having one acute angle equal to 30°. Measure to two decimal places the side a, oppo site the 30° angle, ta: and the side b, adjacent to the 30° PA angle. Find the ratio b Fig. 177 ing a by b to two decimal places. The results found by the pupils in the class should agree. For according to $84 all these triangles, if constructed exactly, are similar to each other. They are therefore really the same triangle drawn to different scales. s by divid 95. Table of tangents. We have seen (894) that for all right triangles having one acute angle equal to 30°, the ratio of the side opposite the 30° angle, to the side adjacent to the 30° angle is the same. We may now draw other right triangles with acute angles of various sizes and make a table which states for each acute angle the ratio of the side opposite to the side adjacent. Such a table is called a table of tangents, and the ratios are called tangent ratios. The TABLE OF TANGENTS OF ANGLES FROM 0° TO 89° 1.732 1.804 1.881 1.963 2.050 2.145 2.246 2.356 2.475 2.605 2.747 2.904 3.078 3.271 3.487 3.732 4.011 4.331 4.705 5.145 5.671 6.314 7.115 8.144 9.514 11.430 14.301 19.081 28.636 57.290 45 75 15 16 .213 .231 .249 .268 .287 .306 .325 .344 .364 .384 .404 46 17 76 77 78 79 18 1.111 19 1.150 20 80 21 81 22 47 48 49 50 51 52 53 54 55 56 57 84 1.235 1.280 1.327 1.376 1.428 1.483 1.540 1.600 1.664 85 27 88 89 |