Find the area of a pennant if a = 18 in., b=6 in., c=6 in., and d= 12 in. 10. Figure 109 represents the end of an "I-beam," a form of steel beam used in building. Make a formula for finding the area of this figure. Find the area if a = 12 in., c=2 in., and b=4.5 in. 11. A concrete porch floor has the form of Figure 110. Make a formula for finding the area of this figure. Find the cost of laying this floor at 17¢ a square foot if a=30 ft., b=20 ft., c=51 ft., and d=7 ft. a FIG. 109. Td FIG. 110. 12. Manila paper comes in sheets 24 in. by 36 in., and costs 3¢ a sheet. The printer cuts it into sheets 12 in. by 18 in. What must he ask for 100 of these smaller sheets so as to gain 25% on the cost? 13. A garden is 30 ft. wide and 40 ft. long. The middle points of the opposite sides are joined by paths 2 ft. wide. Find the area that remains for cultivation. 15. A ranchman has enough barbed wire to build 4 miles of fence. He wishes to inclose with it as large a rectangular field as possible. How long and how wide should he make the field? 16. A half-section of land is 160 rd. by 130 rd. It is divided into fields 40 rd. by 80 rd. by cross fences running parallel to the sides. Find the length of these cross fences. 17. The members of an arithmetic class wish to measure the school pond which is represented in Figure 112. They measure the line AD which is 265 ft. long. DC and AB are measured perpendicular to AD. DC=265 ft., and AB=402 ft. They decide that they can find the area of the pond as accurately as they desire if they regard EAF,GIH, LBM, NCO, and PDQ as triangles and subtract the sum of their areas from the area of ABCD. E D РО N M Y A F G J HL B IJ is the altitude of GIH, BY is the altitude of LBM, and CX is the altitude of NCO. It is found by measuring that EA 98 ft., and AF=82.5 ft.; GH = 130 ft. and IJ=51 ft.; LM-90 ft. and BY=64.2 ft.; NO=192 ft. and CX=57.3 ft.; PD=112 ft. and DQ=73 ft. Find the area of the pond in square feet; also in acres to the nearest .01 acre. = 103. The area of a circle. 1. We first found the area of what kind of plane surface? 2. In finding the area of a parallelogram into what kind of figure was it changed? We shall now find the area of a circle by first changing it to a form that closely resembles a figure whose area we know FIG. 113. www FIG. 114. how to find. Cut a circle into 12 equal sectors by drawing radii as in Figure 113, and fit them together as in Figure 114. 3. What figure does Figure 114 resemble? 4. How does it differ from a parallelogram? 5. What line of the circle is the base of Figure 114? What line of the circle is its altitude? 6. If Figure 114 you find its area? the area of a circle? were exactly a parallelogram how could What rule does this suggest for finding The answers to the above questions suggest the following rule which is proved in geometry: Rule. The area of a circle equals one-half the product of the radius and the circumference. This rule is more easily remembered in the formula Arc, in which A represents the area, r the radius, and c the circumference of the circle. Since c=2πr, then A=rX2πr=πr2. This formula A=r2 is the one most commonly used in finding the area of a circle. It is sometimes stated as the Rule. The area of a circle equals π times the square of the radius. Exercise 95 Find the areas of circles having the following radii. Use π=34. 11. Find the areas of circles with radii 1 in., 10 in., 50 in., 100 in., using both values of T. By how many square inches do the areas differ in each case? If we wish the answer correct to the nearest square inch, for which values of the radii may we use π = 34, and for which must we use π=3.1416? 12. A horse is tethered by a rope 20 ft. long. Over how large an area can he graze? Find the areas of circles having the following diameters. Use T3.1416. Find the areas of circles having the following circumferIn the first three use π=34. In the next three use ences. 25. A circular race track is of a mile in circumference. What is its radius and its area? 26. What is the area of the largest circle that can be cut from a 10-inch square? The area of this circle is what per cent of the area of the square? 27. What part of Figure 115 represents the square on the diameter? What represents the square on the radius? The area of the circle is how many times the area of the square AEOF? The area of the circle is what part of the square ABCD? D E FIG. 115. B 28. In Figure 115 AB is 20 in. Find the area of the square on the diameter. Find the area of the square on the radius. Find the area of the circle. Check your answers to the last two questions in problem 27 by using the areas you have just found. 29. State in words the meaning of the formula c = 2 πr. 30. State in words the meaning of the formula A =πr2. с 31. State in words the meaning of the formula A = r. REVIEW PROBLEMS Exercise 96. General Review 1. Mrs. Simpson has been buying flour in 3 lb. packages which cost 20¢ each. She decides to buy a bag of 24 lb. for $1.25. Which is cheaper, and how much cheaper for 24 lb.? 2. One store advertises 3 lb. of sugar for 25¢, and another 5 lb. for 35¢. Which is cheaper? 3. There are 17 pupils in a cooking class and each needs 2 cups of water for a cooking lesson. How many quarts must there be in a kettle to supply all the class at one time, allowing 4 cups for a quart? 4. An automobile uses 1 qt. of oil every 100 miles and 1 gal. of gasoline every 18 miles. Oil costs 55¢ a gallon and gasoline 23¢ a gallon. Find the cost of oil and gasoline per mile, correct to .001. Find the cost of oil and gasoline for 100 miles; for 1000 miles. 5. A man buys a new automobile in the spring and drives it 5286 miles the first season. At the end of the season he wishes to compute his expenses. He estimates that the machine has depreciated $300 in value. He has worn out 4 tires which cost $23.40 each. He estimates that he has used 1 qt. of oil every 90 miles and 1 gal. of gasoline every 18 miles. Oil has cost him 58¢ a gallon and gasoline 22¢ a gallon. Repairs have cost $9.60. Find the total expenses. Find the expense per mile. If the automobile carried on the average four passengers, was the cost of transportation of these four persons more or less expensive than traveling on the railroad at 2¢ a mile? How much? 6. A man owns of a stock of goods and sells of his share. What part of the stock does he sell? What part does he keep? |