36. The weight of a copper coin varies directly as its thickness and also as the square of its diameter. A coin weighing 11 grams has a diameter of 3 cm. and a thickness of .2 cm. What is the weight when the diameter is 2.5 cm. and the thickness .3 cm.? 37. A's rate of working is to B's as 3:4; B's rate of working is to C's as 6:5. How long will it take C to do work which A can do in 8 hours? 38. The distance through which a body falls from rest varies as the square of the time of falling. If a body falls 257.6 ft. in 4 seconds, how far will it fall in 5 seconds? 39. If a stone weighing 5 pounds falls from rest through a distance of 144.9 ft. in 3 seconds, how far will a stone fall from rest in 3 seconds, if its weight is 500 pounds? See Ex. 38. 40. The tractive force or pull necessary to move a vehicle at a uniform rate (say 3 miles an hour) varies directly as the pressure (weight of vehicle and load) and inversely as the square root of the average radius of the wheels. If on level, paved, or macadam roads a pull of 57 pounds per ton of pressure is necessary when the front and rear wheels average 50 inches in diameter, what pull is necessary to move 3500 pounds when the wheels of the vehicle average 38 in. in diameter ? 41. If a tractive force of 75 lb. is necessary to keep a load of 1 ton in motion in a vehicle whose wheels average 38 in. in diameter, on a dry and hard earth road, what is the tractive force for 5 tons when the wheels average only 26 in. in diameter? See Ex. 40. 42. If on an earth road, in sticky mud in. deep, the tractive force per ton is 119 lb. when wheels 38 in. in diameter are used, what tractive force is required to move 4 tons when the diameters are 50 in. ? 43. The tractive force per ton over dry, cloddy plowed ground is 252 lb. for wheels 50 in. in diameter; what is the tractive force for of a ton when the wheels are 26 in. in diameter ? GRAPHS EXHIBITING EMPIRICAL DATA 106. 1. By a certain plan of life insurance a single premium is paid in order that the insured may receive $100 when he is 60 years old or that his beneficiaries may receive $100 if he dies before that age. The premium depends upon the age at which the insurance is taken. Age next birthday : 35 40 45 15 20 25 30 50 Premium: $41.20 $45.25 $49.30 $53.80 $59.10 $65.10 $72.05 $80.20 Draw a graph and use it to find the premium one would pay at the age of 18, 23, 32, 38, 43. It is convenient to mark off along the x-axis the ages above 15; along the y-axis the premiums above $40. By this device a much smaller sheet of square paper can be used. See Fig. 14. 2. In a city in the northern part of the United States the times of sunrise on Ages above 15 years FIG. 14. certain days in May, June, and July are as follows: Draw a graph showing the hour of sunrise from May 10 to July 30. Mark off on one axis the number of days after May 10, on the other axis the number of minutes after 3: 44 A.M. the exponent L is called the logarithm of the number N, to the base 10. For example, 102 100, hence 2 is the logarithm of 100, to the base 10. 108 = We write, 2 = log 100. 1000, hence 3 is the logarithm of 1000, to the base 10. 10 = 3.16227+ (verify this), hence .5 is the logarithm of 3.16227+, to the base 10. We write, .5= log 3.16227+. We write, 1.5 = log 31.6227+. 10 101(10)= 31.6227+, hence 1.5 is the logarithm of 31.6227+, = the base 10. to As in this chapter all logarithms are taken to the same base 10, no confusion will arise from the omission, hereafter, of the phrase "to the base 10." Verify the following statements: 108. On a piece of square paper as small as that in Fig. 15, it is not convenient to draw a curve that will exhibit the logarithms of numbers ranging from .001, the smallest, to 1000, the largest number considered above. The curve shows the logarithms of positive numbers between .1 and 50. The numbers are laid off on the x-axis, their respective logarithms on the y-axis. 1 = 1.5 log 10 locates the point D log 31.6+ locates the point E In drawing the curve, some additional values were used. ORAL EXERCISES ON THE LOGARITHMIC CURVE 109. 1. Where does the curve cut the x-axis? How much is log 1? 2. What is the algebraic sign of the logarithms of all numbers larger than 1? 3. For what range of numbers are the logarithms negative? 4. Does the logarithmic curve extend to the left of the y-axis? 5. Do negative numbers have real logarithms? 6. Does the logarithm increase as a variable number increases ? By inspection of Fig. 15, find approximately the logarithms of the following numbers: By inspection of Fig. 15, find approximately the numbers corresponding to the following logarithms: Find by inspection, how many times greater 28. log 4 is than log 2. 29. log 8 is than log 2. 30. log 16 is than log 2. 31. log 32 is than log 2. 32. log 9 is than log 3. 33. log 27 is than log 3. |