13. The volume of a right circular cone varies jointly as its altitude and the square of the radius of its base. If the volume of a cone 7 inches high and radius of base 6 inches, is 264 cubic inches, find the volume of a cone 5 inches high and base with radius 2 inches.

14. Find the radius of a sphere whose volume is the sum of the volumes of two spheres whose radii are 3 feet and 5 feet respectively.

15. If 100 men can do a piece of work in 20 days, how long will it take 40 men to do it?

16. The weight of a body near the earth varies inversely as the square of its distance from the center of the earth. If the radius of the earth is 4000 miles, at what distance from the earth would a body weigh only half as much?

17. The illumination from a source of light varies inversely as the square of the distance from the source. How far must a photographic plate, that is 12 inches from a lamp, be moved to receive twice as much light?

18. The number of times a pendulum oscillates in a given time varies inversely as the square root of its length. If a pendulum 39.1 inches long oscillates once a second, how long must a pendulum be to oscillate three times a second?

19. The velocity acquired by a body falling from rest varies as the square root of the height. If a body, that fell from a height of 400 feet, acquires a velocity of 160 ft. per second, with what velocity will a body strike the earth after falling from a height of 1000 feet?

20. The heat loss in an electric circuit varies directly as the intensity of the current and the square of the resistance. If the heat loss is 250 watts when the current is 10 amperes and the resistance 5 ohms, find the heat loss when the current is 20 amperes and the resistance 1 ohm.

21. The variation of volume and pressure of a gas is expressed by the formula P1V1 = P2V 2.

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A certain mass of gas has a volume of 5000 liters under a

pressure of one atmosphere. Find to how many atmospheres the pressure must be increased to reduce the volume to 1000 liters.

22. A steel tank having a capacity of 4 cubic feet is filled with oxygen under a pressure of 10 atmospheres. Find how much gas is in the tank measured at standard atmospheric pressure.

23. Two electric lights of 16 and 64 candle power, respectively, are placed in front of a picture so as to illuminate it equally. The 16 candle power lamp is placed 10 feet from the picture. At what distance must the other be placed?

REVIEW EXERCISE XXIII

1. If the illumination of an object at a distance of 10 feet from a source of light is 2, what is the illumination at 40 feet? (Illumination varies inversely as the square of the distance.)

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2. If x varies jointly as y and z, and inversely as the square of and if x = ω, 30 when y 3, = 2 5 and w = 4, what is the value of x when y 9, 2= 12, and w = 6?

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3. The volume of gas varies as the temperature and inversely as the pressure. If the volume is 200 cubic inches when the temperature is 260° and the pressure is 15, what is the volume when the temperature is 390° and the pressure

5. The pressure of wind on a plane surface varies jointly as the area of surface, and the square of the wind's velocity. The pressure on a square foot is 1 pound when the wind is moving 15 miles per hour; find the velocity of the wind when the pressure on a square yard is 16 pounds.

6. If y is the sum of two quantities which vary directly as x2 and inversely as a respectively, and y = - when x = 1, and y 37 when x=-2, find y when x = -. 一号

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7. The distance fallen by an object from a position of rest

varies as the square of the time of falling. If a body falls 144 feet in 3 seconds, how far will it fall in 10 seconds?

8. The electrical resistance of a wire varies directly as the length and inversely as the area of the cross section. If the resistance of 130 yds. of wire in. in diameter is 1 ohm, what is the resistance of 100 yds. of 1⁄2 inch wire?

9. Given that the area of a triangle varies jointly as its base and altitude, what will be the base of a triangle whose altitude is 12, equivalent to the sum of two triangles whose bases are 10 and 6, and altitudes 3 and 9 respectively?

10. If the area of a circle varies as the square of its radius, find the radius of a circle equivalent to the sum of two circles whose radii are 5 and 12 respectively.

11. The weight of a body varies inversely as the square of its distance from the center of the earth. If a body weighs 30 lbs. on the surface of the earth (4000 miles from the center), what would be the weight of the body at a distance of 24000 miles from the surface of the earth?

12. The amount of heat received from a stove varies inversely as the square of the distance from it. A person sitting 16 feet from a stove moves up to 4 feet from it. How much will this increase the amount of heat received? 13. Given that y equals the sum of three quantities which vary as x, x2, and 3 respectively. When x = 1, y = 4; x3 when x = 2, y 8; when x 3, y= 18. Express y in

terms of x.

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14. The intensity of illumination from a source of light varies inversely as the square of the distance from the source. A certain incandescent light, when placed at a distance of 49 centimeters from a screen, is found to give the same illumination as a "standard" 16 candle power lamp placed at a distance of 51 centimeters from the screen. What is the candle power of the incandescent light?

15. Three spheres of lead, whose radii are 6, 8, and 10 inches respectively, are united into one. Find the radius of the resulting sphere, if the volume of a sphere varies as the cube of its radius.

16. The area of an ellipse varies jointly as its axes. The area of an ellipse is 110 square inches when the axes are 14 inches and 10 inches. Find the area when the axes are 8 and 10 inches.

17. The square of the time of a planet's revolution about the sun varies as the cube of its distance from the sun. Neptune is about thirty times as far from the sun as the earth; find the length of his year compared with ours.

CHAPTER XXIV

PROGRESSIONS

172. A series is a succession of numbers, each of which, after the first, is derived from the preceding number or numbers by some fixed law.

The successive numbers are called the terms of the series. The first and last terms are called the extremes, and all the others, the means.

Thus, the series 2, 4, 6, 8, 10 is formed by adding 2 to each term to obtain the succeeding term.

ARITHMETICAL PROGRESSION

173. A series, each term of which, after the first, is derived from the preceding term by the addition of a constant number, is called an arithmetical series or arithmetical progression. The constant number, which is added to each term to obtain the next term, is called the common difference, since the difference between any term and its preceding term is constant.

Thus the series above, 2, 4, 6, 8, 10 is an arithmetical series and the common difference is 2.

A. P. is the usual abbreviation for the words arithmetical progression.

If the common difference is a positive number, the series is increasing, but if the difference is a negative number, the series is decreasing.

Note that the common difference is always found by subtracting one term from the succeeding term.

If a represents the first term of an A. P. and d, the common difference, the general form of such a series is, a, a + d,