and both diameter and height of the gallon measure are 7.068 inches. Ex. 1. Calculate the height and diameter of a standard litre measure whose capacity is 1000 c.cm., and whose height is twice its diameter. 22. Compare the method of extracting a cube root by logarithm tables with that by means of the graph of x3 (chapter VI), and with the ordinary arithmetical method. The graph method is only a rough method. The arithmetical extraction of a square root is cumbersome, for a cube root it is forbidding, and for higher roots the labour is so great that to calculate such a number as 2.7183.142 without logarithm tables is possibly beyond the patience of any man. Logarithm tables are a great help for multiplication and division, but their true value as a labour-saving device first appears in root extraction. Ex. 2. Calculate the number 2.7183.142. 23. Can you generalize the relation log (dxd xd) = log d+log d+log d, or (more compactly) log d3 = 3 logd ?-If n is any whole number log an = n log d, for this is only the short way of writing log (dxdxd ...) log d+log d+ = 103k = = = = there being n factors d on the left, and n terms logd on the right. Can you deduce this relation from the meaning of logarithm?-Suppose log dk, then d 10k, and d3 = (10)3 so that log d3 3k, that is, 3 log d. And so for the case when 3 is replaced by n. Is it true that log dn Put log dk, then d integral or fractional dn nk = n log d. = = n log d = if n is fractional ?10k and no matter whether n is (10k)n = 10kn, so that log d = 24. The quantities z and u being related by * = 10", look out a number of corresponding values in the tables and draw the graph of z as a function of u from z = 1 to 2 = 10. What are the corresponding limits for u? It was stated (in art. 1 of this chapter) that the graphs y = 1.01* and 1.001 differed only in the graduation of the x-axis. Can you choose such a scale for u that the curve z 10u y = = will fit the curve y = 1.01.- runs from 0 to 1, and x runs from 0 to 231, so let us try making x = 231 × u for all values of x and u. Then for any value of x (1.01231); and since 1.01231 1.01x y = = 1.01231u this is = 10. Therefore y = ≈; and to make ≈ all values of x and u does make the graphs fit. = 25. Can you explain why our logarithm table was useful to give x from y or y from x when y 1.0023? That is, how would you make the curve z = 10u fit the curve 1.0023?- -x runs from 0 to 1000, u from 0 to 1. If 1000u for all values of u, then y (1.00231000), and this is = 10" since 1.00231000 x from the table we multiplied the logarithm there given by 1000, so that the relation x 1000u expresses exactly what we did. = = Ex. 3. Draw the graph of z 10", not only from u = 0 to u = 1, but for as great a range of values of u as possible. EXERCISES. 4. In t seconds a body falls 16t2 feet. How many feet does it fall in 5 seconds, and how many seconds does it take to fall 100 feet? 5. From a cylinder of wood with a diameter of 11 inches a rectangular beam is cut. If the breadth of the beam is 8 inches, find the depth from the formula d ✔D2-b2, where bis the breadth and d the depth of the beam and D the diameter of the cylinder. If the strength of a beam varies as bd2, compare the strengths of beams 3, 4, 5, 6, 7, 8 inches broad cut from a cylinder 11 inches in diameter, and so find, roughly, the breadth of the strongest beam which can be cut from the cylinder. 6. The Board of Trade rule for flat stayed surfaces of marine boilers is p: c (16 t + 1)2 is the safe working pressure s-6 in lbs. per square inch, t is the thickness of the plate in inches, and s is the area of plate supported by each stay in square inches. The pitch' of the stays, a, is given by t steel plates. Assume that s = 1.5a2, c = 60, and find the safe working pressure which may be used with plates 1 inch thick. 7. The height of a pint pot of a certain shape is 12.6 cm. to the nearest tenth. A litre being 1.76 pints to the nearest hundredth, find the heights of pots of the same shape that will hold 3 decilitres, 5 decilitres, and 1 litre. 8. The population of the United Kingdom was estimated at 37,802,000 in the middle of the year 1891, and at 41,551,000 ten years later. Assuming that population increases (i) by the same number of individuals every year, (ii) by a constant percentage every year, estimate the population in the middle of the years 1894 and 1897. 9. The earth completes a revolution round the sun in 365 days, and the planet Venus completes a revolution in 225 days. The distance from the earth to the sun is 90 million miles. Find the distance of Venus from the sun, assuming that the square of the time of a planet's revolution is proportional to the cube of its distance from the sun. 10. Obtain the quotients (y2-3y+2)-(x2-3x+2) y- - x and y5 - x5 The limiting values of these quotients when y is put equal to x after division are called the differential coefficients of x2-3x+2 and 5 respectively. Show that the differential coefficient of x2-3x+2 is 2x-3 and that of x is 5x4. 11. A ladder 7.9 metres long is placed against a wall, the foot of the ladder being 3.8 metres from the bottom of the wall. If the height of the point on the wall reached by the ladder is x metres, find x, given that x2 +3.82 = 7·92. 12. The area of the surface of the human body is proportional to the square of the height, and the weight is proportional to the cube of the height. The loss of heat by the body varies as the surface area, and this loss must be made up by food. Does a man of 150 lbs. or his son of 110 lbs. need more food for this purpose in proportion to his weight? If the father needs 7 ounces a day for this purpose, what does the son need? 13. Given that the area of any triangle the lengths of whose sides are a, b, and c is√s (s - a) (s – b) (s - c), where s = (a+b+c), reduce this to its simplest form for a triangle in which a2 b2+ c2, and calculate the area when b 6.78 inches, c = = = 4.39 inches. 14. The gas-service pipe to a house 75 feet from the main is 7/8 inch in diameter. For how many burners, each taking 5 cubic feet of gas per hour, will this serve? The number of cubic feet per hour delivered by a pipe on that main is 1000 d5 0.45 L where d is the diameter of the pipe in inches, and L is the length of the pipe in yards. 15. A penny measures 1.23 inches across and a halfpenny 1 inch. If the thicknesses were in the same ratio, what fraction (as a decimal to the nearest hundredth) would the weight of the halfpenny be of the weight of the penny? If a penny weighed twice as much as a halfpenny, and was of the same thickness, what would it measure across (to the nearest hundredth of an inch), the halfpenny measuring 1 inch? If a disc measures a inches across and b inches in thickness, then its volume is 0·7854 × a×a× b cubic inches. 16. The time of swing of a pendulum is proportional to the square root of its length. If a pendulum 39 inches long swings once per second, how many swings per minute will be made by a pendulum 5 feet long? 17. On a map drawn on a scale of an inch to a mile two points are found to be 1.3 inches apart. One point is at the sea level, the other at a height of 1,720 feet. Lay off on squared paper to any suitable scale the horizontal and vertical distances between these points, and by measurement find the length of wire that would run straight between these points. Having given that (distance between the points)2= (horizontal distance)2 +(vertical distance)2, calculate the length of wire in feet to the nearest hundred. 18. In return for an immediate payment of £575 17s. 6d. a Life Assurance Society undertakes to pay £1,000 at my death, reckoning that on the average a man of my age will live nineteen years more. Find, to the nearest tenth per cent, what rate of interest the Society pays on the average for money thus deposited with it, assuming that a sum of £p invested at r per cent. amounts in n years to £p (1+r/100)”. Investigate the correctness of the formula used. 19. Find the value of (a1/5+b1/6)1/10÷c1/4, where a = 1009'6, b = 2419.3, c = 5769. 20. Explain the meaning of a5 and a1/5. Between what integers does 871/5 lie? Give reasons for your answer. 21. The formula p = P÷2·7180/26000 gives the pressure p of the atmosphere in pounds per square inc at a height of a feet above sea level, P being the pressure in pounds per square inch at the sea level. Find the pressure at the top of Snowdon, 3570 feet above sea level, when P = 15. 22. Give an account in continuous English of indices, explaining the convenience of the notation for the integral index, and the extension from integral to negative and fractional indices. Importance is attached to the form as well as the matter of the account. 23. A jet of water issues from the side of a tank with a horizontal velocity of √2gH feet per second, where H feet is the head of water above the orifice, and g a certain constant. The water retains this horizontal velocity unchanged while falling, and in t seconds from issuing it falls 0·5 gt2 feet. Taking H = 10, and g = 32, find whereabouts the jet will strike a horizontal plane 100 feet below the orifice. 24. The relation between p and v indicated by the table = is suspected to be given by the equation p = k, where x and k are constants. Allowing for slight errors of observation, find, as nearly as you can, the values of x and k, and give the value of p corresponding to the value 6'4 of v. 25. If a* = b2, and b = ao, show that y=x/c. With the aid of the tables of logarithms apply this result to find the values of loge 5 and log, 211, where e 2.718. 26. It is said that any angle ABC can be trisected by the following method: On tracing-paper draw a circle, centre O. Produce any radius OD to E so that DE = OD, and through D draw MN of indefinite length perpendicular to OE. Place the tracing-paper figure so that the whole of the circle lies within the angle ABC, arrange it so that MN passes through B, one arm of the angle passes through E and the other touches the circle. Then prick through D and O, and join the points thus obtained to B.' Carry out this construction for an angle of 80°, using a circle with a radius of 3 cm. ; test the result with your protractor, and either justify or disprove the soundness of the method. 27. The weight of the model of a bell made of the same metal as the bell on the scale of 5 centimetres to the metre is 26 grams. What is the weight of the bell? 28. Draw a circle of radius 6.6 cm. Draw a diameter AB and produce it to C so that BC= 3.2 cm. Suppose a line originally lying along CBA to turn about C till it ceases to cut the circle. Call P and Q its points of intersection with the circle. Make a table |