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594. Similar Figures. Figures that have the same shape are called similar figures.
595. The corresponding lines of similar figures are proportional.
596. The surfaces of similar figures are to each other as the squares of their corresponding dimensions; and their volumes are to each other as the cubes of their corresponding dimensions.
597. The corresponding dimensions of similar figures are to each other as the square roots of their surfaces, or as the cuhe roots of their volumes.
598. Examples. 1. A rectangle is 8 in. long and 6 in. broad. Find the length and the area of a similar rectangle whose breadth is 9 in.
Solution. 6:9 = 8 in.: required length.
2. The altitude of a right prism that contains 8 cu. ft. is 3 ft. Find the altitude of a similar right prism that contains 27 cu. ft.
Solution. 3 ft.: required altitude = V5 : V27 = 2:3.
1. If the diameter of the moon is reckoned at 2000 mi., and that of the earth at 8000 mi., find the ratio of their surfaces and the ratio of their volumes.
2. If the diameters of two circles are 20 in. and 40 in., find the ratio of their circumferences, and of their surfaces.
3. If the areas of two circles are 8000 sq. in. and 36,000 sq. in., respectively, find the ratio of their diameters.
4. If the volumes of two spheres are 100 cu. in. and 1000 cu. in., respectively, find the ratio of their diameters.
5. If an ox 7 ft. in girth weighs 1500 lb., what will be the girth of a similar ox that weighs 2500 lb.?
6. The surface of a pyramid is 560 sq. in. What is the surface of a similar pyramid whose volume is 27 times as great?
7. The volume of a pyramid is 1331 cu. in. What is the volume of a similar pyramid whose surface is 4 times as great?
8. If a well-proportioned man 5 ft. 10 in. high weighs 160 lb., what should a man 6 ft. high weigh, to the nearest tenth of a pound? What should be the height, to the nearest tenth of an inch, of a man who weighs 210 lb.?
9. A three-gallon jug and a one-gallon jug are similar. Find to three decimals the ratio of their diameters.
10. Two hills have exactly the same shape; one is 900 ft. high, the other 1200 ft. Find the ratio of their surfaces, and also the ratio of their volumes.
11. A ball 3 in. in diameter weighs 4 lb.; another ball of the same metal weighs 9 lb. Find the diameter of the second ball to the nearest thousandth of an inch.
12. If Apollo's altar were a perfect cube 10 ft. on an edge, what would be the edge of a new cubical altar containing twice as much stone?
13. A man standing 40 ft. from a building 24 ft. wide observed that, when he closed one eye, the width of the building hid from view 90 rods of fence which was parallel to the width of the building. Find the distance from the eye of the observer to the fence.
14. A bushel measure and a peck measure are of the same shape! Find the ratio of their heights.
15. If the height and the diameter of a cylinder are both doubled, in what ratio is the volume altered?
599. As in decimals we often require a result accurate to a specified number of places, so in common fractions we often require the most nearly accurate value of a ratio that can be expressed by a fraction with a denominator limited to a' certain size.
600. Example. Find the most nearly accurate value of the ratio of the circumference of a circle to the diameter expressed by a fraction with a denominator less than 10; less than 100; less than 1000.
Solution. The ratio 3.1416 is true to the nearest ten-thousandth. Reducing TWA t0 'te lowest terms, we have TVj^.
Then, as in the margin, we divide the denom177) 1250 (7 inator by the numerator; the last divisor by
1239 the last remainder; and so on, as in rinding
ll)l77(16 the greatest common measure.
176 If, therefore, we divide both terms of the
fraction TWj by the numerator, we have;
11 "' 'ttt
and if we omit the fraction in the denominator, we have for the required ratio with a denominator less than 10, 31, or V.
If we put the fraction Ty7 in the form of r-j-r- and omit the fraction
in the denominator, the ratio becomes
a -i— = Q 1 6 = 3 5 5 .
which shows that 272- is the most nearly accurate value of the ratio expressed by a fraction with a denominator less than 100, and that § Tf is the most nearly accurate value of the ratio expressed by a fraction with a denominator less than 1000.
601. Continued Fractions. After the quotients have been found the results may be written in a fractional form as follows:
3 + -U
Such a fraction is called a continued fraction.
602. To find the successive approximate values of a continued fraction we begin at the top and take first one, then two, then three, and so on, of its parts. Thus,
The first approximate value is 8.
The second is 3 + I = 272-.
The third is 3 + —^ = 3 + ^j = f f f •
and this = 3 + =4r= 3 + tffr = fHJ, or 3.1416.
603. In reducing the part of a continued fraction selected for an approximate value, we begin with the last fraction.
1. Change JT, \%, fgj, \3?* to continued fractions.
2. Find the approximate values of %%; -f^; |ff.
3. Find a series of fractions approximating to 0.236; 0.2361; 1.609.
4. Find a series of fractions approximating to 0.382; 1.732; 0.6253.
5. Find approximate values of \J|; f Jf; \\%; f \J.
6. Find the proper fraction that, when changed to a continued fraction, will have 2, 3, 5, 6, 7 as quotients.
7. Find a series of fractions approximating to the ratio of the pound troy (5760 gr.) to the pound avoirdupois (7000 gr.).
8. Find a series of fractions approximating to the ratio of the side of a square to its diagonal; that ratio being 1:1.414214, nearly.
9. Find a series of fractions approximating to the ratio of the ar to the square chain, from the equality
1 ar = 0.2471 sq. ch.
10. Find a series of fractions approximating to the ratio of the weight of the 48-pound shot to the weight of the French shot of 24kg.
11. If the mean diameter of the Earth is reckoned at 7912 mi., and that of Mars 4189 mi., find a series of fractions approximating to the ratio of the mean diameters of these two planets.
12. Find a series of fractions approximating to the ratio of a cubic yard to a cubic meter, from the equality
1 cu. yd. = 0.76453cbm.
13. Find a series of fractions approximating to the ratio of the kilometer to the mile, from the equality
lm = 1.09362 yd.