. Railroad construction. Theory and practice . The area may thenbe obtained independent of the center depth as follows: Let b s = the slope ratio of the side slopes = cot ^ = 2a (See Fig. 50.) Then the Area ^1 /xi-\-Xr\ 2\ s J (xi-\-Xr) Xr Xr XI XI s 2 ab_2 XIX, ~2 (54) 86 RAILROAD CONSTRUCTION. §77. The true volume, according to the prismoidal formula, of alength of the road measured in this way will be r xix/ ah /xi + xi x/ + x/ 1 ah\L s 2^ V 2 2 s 2 / x^x/ ah~2 If computed by averaging end areas, the approximate volumewill be Fj^/a:/ ab xixr ah\. Subtracting this result from the true volume
. Railroad construction. Theory and practice . The area may thenbe obtained independent of the center depth as follows: Let b s = the slope ratio of the side slopes = cot ^ = 2a (See Fig. 50.) Then the Area ^1 /xi-\-Xr\ 2\ s J (xi-\-Xr) Xr Xr XI XI s 2 ab_2 XIX, ~2 (54) 86 RAILROAD CONSTRUCTION. §77. The true volume, according to the prismoidal formula, of alength of the road measured in this way will be r xix/ ah /xi + xi x/ + x/ 1 ah\L s 2^ V 2 2 s 2 / x^x/ ah~2 If computed by averaging end areas, the approximate volumewill be Fj^/a:/ ab xixr ah\. Subtracting this result from the true volume, we obtain as thecorrection Correction =— (x/—a:/) (^/—:i:/0 {^^) This shows that if the side distances to either the right orleft are equal at adjacent stations the correction is zero, andalso that if the difference is small the correction is also smalland very probably within the limit of accuracy obtainable bythat method of cross-sectioning. In fact, as has already beenshown in the latter part of § 75, it will usually be a useless. Fig. 50. refinement to compute the prismoidal correction when themethod of cross-sectioning is as rough and approximate as thismethod generally is. 77. Equivalent level sections. These sloping two-level sections are sometimes transformed into level sections of equal § 77. EARTHWORK. 87 area, and the volume computed by the method of level sections(§ 74). But the true volume of a prismoid with sloping endsdoes not agree with that of a prismoid with equivalent bases andlevel ends except under special conditions, and when this methodis used a correction must be applied if accuracy is desired,although, as intimated before, the assumption that the sectionshave uniform slopes will frequently introduce greater inaccuraciesthan that of this method of computation. The following dem-onstration is therefore given to show the scope and limitationsof the errors involved in this much used method. In Fig. 50, let d^ be the center height which gives an
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