CHAPTER XXIII VARIATION 228. General principles. The number x is said to vary directly as the number y when the ratio of x to y is constant. This we symbolize by where k is a constant. x = xxy, or k, y (1) Thus if a man walks at a uniform speed, the distance that he goes varies directly as the time. If the length of the altitude of a triangle is given, the area of the triangle varies directly as the base. The volume of a sphere varies directly as the cube of its radius. The number x is said to vary inversely as the number y when varies directly as the reciprocal of y. Thus x varies inversely as y when where k is a constant. Thus the speed of a horse might vary inversely as the weight of his load. The length of time to do a piece of work might vary inversely as the number of laborers employed. The intensity of a light varies inversely as the square of the distance of the light from the point of observation. If represents the intensity of light and d the distance of the light from the point of observation, we have The number x is said to vary jointly as y and z when it varies directly as the product of y and z. Thus x varies jointly as x and when Thus a man's wages might vary jointly as the number of days and the number of hours per day that he worked. 2 The number x is said to vary directly as y and inversely as z when it varies directly with. Thus the force of the attraction of gravitation between two bodies varies directly as their masses and inversely as the squares of their distances. If m represents the masses of two bodies, d their distance, and G the force of their attraction due to gravity, then = 1. If a varies inversely as the square of b, and if a 2 when b = 3, what is the value of a when bis 18? 2. The volume of a sphere varies as the cube of its radius. A sphere of radius 1 has a volume 4.19. What is the volume of a sphere of radius 3? Solution: Let V represent the volume and r the radius of the sphere. Then by (1), Determine k by substituting, Then V k. 3. If xy xx+y, and x = 1 when y = 1, find x when y = 8. 4. The area of a circle varies as the square of the radius. If a circle of radius 1 has an area 3.14, find the area of a circle whose radius is 21. 5. Find the volume of a sphere whose radius is .2. HINT. See exercise 2. 6. The volume of a circular cylinder varies jointly with the altitude and the square of the radius of the base. are each 1 has a volume of 3.14. altitude is 15 and whose radius is 3. A cylinder whose altitude and radius Find the volume of a cylinder whose 7. The weight of a body of a given material varies directly with its volume. If a sphere of radius 1 inch weighs of a pound, how much would a ball of the same material weigh whose radius is 16 inches ? 8. The distance fallen by an object starting from rest varies as the square of the time of falling. If a body falls 16 feet in 1 second, how far will it fall in 6 seconds? 9. A body falls from the top to the bottom of a cliff in 31 seconds. How high is the cliff? 10. A triangle varies in area jointly as its base and altitude. The area of a triangle whose base and altitude are each 1 is. What is the area of a triangle whose base is 16 and altitude 7? 11. If 6 men do a piece of work in 10 days, how long will it take 5 men to do it? 12. If 3 men working 8 hours a day can finish a piece of work in 10 days, how many days will 8 men require if they work 9 hours a day? 13. An object is 30 feet from a light. To what point must it be moved in order to receive (a) half as much light, (b) three times as much light? 14. The weights of objects near the earth vary inversely as the squares of their distances from the center of the earth. The radius of the earth is about 4000 miles. If an object weighs 150 pounds on the surface of the earth, how much would it weigh 5000 miles distant from the surface? CHAPTER XXIV PROBABILITY 229. Illustration. If a bag contains 3 white balls and 4 black balls, and 1 ball is taken out at random, what is the chance that the ball drawn will be white? This question we may answer as follows: There are 7 balls in the bag and we are as likely to get one as another. Thus a ball may be drawn in 7 different ways. Of these 7 possible ways 3 will produce a white ball. Thus the chance that the ball drawn will be white is 3 to 7, or 3. The chance that a black ball will be drawn is 4. 230. General statement. It is plain that we may generalize this illustration as follows: If an event may happen in p ways and fail in q ways, each way being equally probable, the chance or probability that it will happen in one of the p ways is Ρ The chance that it will fail is q (1) (2) The sum of the chances of the event's happening and failing is 1, as we observe by adding (1) and (2). The odds in favor of the event are the ratio of the chance of happening to the chance of failure. In this case the odds in favor are EXERCISES 1. If the chance of an event's happening is, what are the odds in its 2. From a pack of 52 cards 3 are missing. What is the chance that they are all of one suit? Solution: The number of combinations of 52 cards taken 3 at a time is 52.51.50 C52, 3 = This represents p+q. The number of combinations of the 1.2.3 13 cards of any one suit taken 3 at a time is C13, 3 = 3. What is the chance of throwing one and only one 6 in a single throw of two dice? Solution: There are 36 possible ways for the two dice to fall. This represents pq. Since a throw of two sixes is excluded there are 5 throws in which each die would be a 6, that is, 10 in all in which a 6 appears. This represents p. 4. A bag contains 8 white and 12 black balls. What is the chance that a ball drawn shall be (a) white, (b) black? 5. A bag contains 4 red, 8 black, and 12 white balls. What is the chance that a ball drawn shall be (a) red, (b) white, (c) not black? 6. In the previous problem, if 3 balls are drawn, what is the chance that (a) all are black, (b) 2 red and 1 white? 7. What is the chance of throwing neither a 3 nor a 4 in a single throw of one die? 8. What is the chance in drawing a card from a pack that it be (a) an ace, (b) a diamond, (c) a face card? |