That is, if one quantity varies inversely as another, the product of the quantities is constant. If 6 men can do a piece of work in 12 hours, 3 men can do the same work in 24 hours, and 1 man in 72 hours, and the products 6 × 12, 3 × 24, 1 × 72 are equal. That is, the number of hours varies inversely as the number of men working. 4. Joint Variation. One quantity is said to vary as two others jointly, when it varies as the product of the others. Thus, if x varies as y and z jointly, then yz k, a constant. For example, the number of miles a train runs varies as the number of hours and the number of miles it runs an hour jointly. It will run 40 miles in 2 hours at a rate of 20 miles an hour, 90 miles in 3 hours at the rate of 30 miles an hour, 120 and 40 90 5. One quantity is said to vary directly as a second and inversely as a third, when it varies as the second and the reciprocal of the third jointly. Thus, if x varies directly as y and inversely as z, then 6. In all the preceding cases of variation, the constant can be determined when any set of corresponding values of the quantities is known. Ex. 1. If xxy, and x = 3 when y = 5, what is the value of the constant? Ex. 2. If x varies inversely as y, and if y = 4 when x=7, find the value of x when y = 12. From xyk, we obtain k Therefore Consequently, when = 28. xy = 28. y = 12, 12 x = 28; whence x = 21. Ex. 3. The volume of a gas varies inversely as the pressure when the temperature is constant. When the pressure is 15, the volume is 20; what is the volume when the pressure is 20? Let v stand for the volume and p for the pressure. Consequently, when p = 20, 20 v = 300; whence v = 15. EXERCISES III. If xxy, what is the expression for x in terms of y, 1. If x = 10 when y = =号? 2. If x a when y = 2 a? 3. If xx y2, and x = 5 when y = 7, what is the expression for x in terms of y? 4. If xxy, and x = 3(a3 + b3) when y = 25(a2 + 2 ab + b2), what is the expression for y in terms of x? 6. If x = 34 when y = }} ? y2' and x = 41⁄2 when y = }, what is the expression for y in 10. If xy, and x = a when ya2, what is the value of y when x = a2b? 11. If xy2, and x = 5 when y = 3, what is the value of x when y = 15? 12. If x∞ √y, and x = a + m when y = (a — m)2, what is the value of x when y = (a + m)1? and x = 3 when y = }, what is the value of x when y = 44? Y' 14. The circumference of a circle whose radius is 6 feet is 37.7 feet. What is the circumference of a circle whose radius is 9.5 feet, if it be known that the circumference varies as the radius? 15. An ox is tied by a rope 20 yards long in the center of a field, and eats all the grass within his reach in 21 days. How many days would it have taken the ox to eat all the grass within his reach if the rope had been 10 yards longer? 16. The volume of a sphere whose radius is 7 inches is 1437.3 cubic inches. What is the volume of a sphere whose radius is 10 inches, if it be known that the volume varies as the cube of the radius ? It has been found by experiment that the distance a body falls from rest varies as the square of the time. 17. If a body falls 256 feet in 4 seconds, how far will it fall in 10 seconds? 18. From what height must a body fall to reach the earth after 15 seconds? It has been found by experiment that the velocity acquired by a body falling from rest varies as the time. 19. If the velocity of a falling body is 160 feet after 5 seconds, what will be the velocity after 8 seconds? 20. How long must a body have been falling to have acquired a velocity of 256 feet? 21. The surface of a cube whose edge is 5 inches is 150 square inches. What is the surface of a cube whose edge is 9 inches, if it be known that the surface varies as the square of its edge? 22. It has been found by experiment that the weight of a body varies inversely as the square of its distance from the center of the earth. If a body weighs 30 pounds on the surface of the earth (approximately 4000 miles from the center), what would be its weight at a distance of 24,000 miles from the surface of the earth? It has been found by experiment that the illumination of an object varies inversely as the square of its distance from the source of light. 23. If the illumination of an object at a distance of 10 feet from a source of light is 2, what is the illumination at a distance of 40 feet? 24. To what distance must an object which is now 10 feet from a source of light be removed in order that it shall receive only one-half as much light? 25. At what distance will a light of intensity 10 give the same illumination as a light of intensity 8 gives at a distance of 50 feet? CHAPTER XXVI. DOCTRINE OF EXPONENTS. 1. We have already abbreviated such products as aa, aaa, ɑɑɑɑ, ......., aua......n factors, by a2, a3, a, ..., a", respectively, and called them the second, third, fourth, ..., nth, powers of a. This definition of the sym bol a" requires the exponent n to be a positive integer. Thus 25 means the product of 5 factors, each equal to 2. But 20 has, as yet, no meaning, since 2 cannot be taken 0 times as a factor. For a similar reason, 2-5, 24, 2√3, and 2-1, are, as yet, meaningless. 2. Nevertheless, having introduced into Algebra the symbol a", it is natural to inquire what it may mean when n is 0, a rational negative or fractional number, an irrational number, etc. We shall find that, by enlarging our conception of powers, quite clear and definite meanings can be given to such expressions as 2o, 3-2, 43, 5v2, 6v−1. 3. The discussion of powers, in general, therefore naturally divides itself into six cases. (1) Powers with positive integral exponents. (2) Powers with zero exponents. (3) Powers with negative integral exponents. (4) Powers with fractional (positive or negative) exponents. (5) Powers with irrational exponents. (6) Powers with imaginary exponents. The consideration of powers with imaginary exponents will be given in Part II., Text-Book of Algebra. Positive Integral Powers. 4. The principles upon which operations with positive integral powers depend have been proved in the preceding chapters. For the sake of emphasis, and for convenience of reference in enlarging our conceptions of powers, we restate them here: aman am+n. am-n, when m>n; (i.) 1 when m<n.• 5. The meaning of a symbol may be defined by assuming that it stands for the result of a definite operation, as was done in letting or by enlarging the meaning of some operation or law which was previously restricted in its application. In the latter way, negative numbers were introduced by extending the meaning of subtraction. 6. We now enlarge the meaning of powers by assuming that the principle That is, the zeroth power of any base, except 0, is equal to 1. E.g., 1o = 1, 5o = 1, 99o =1, (a + b)o = 1, etc. |