Intersection of two straight lines

The value of t used in this equation is chosen at the appropriate P-level and at ( i + 2 - 4) degrees of freedom. The standard deviations in equation (5.20) are calculated on the assumption that the values for the two lines, s y/ i and S(y/ )2, are sufficiently similar to be pooled using an equation analogous to equation (3.3)  

These equations seem formidable, but if a spreadsheet such as Excel is used to obtain the equations of the two lines, the point of intersection can be determined at once. The Syj values can then be pooled, s, etc. calculated, and the confidence limits found using the program s equation-solving capabilities. 

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As a rule, in three-dimensional space the intersection of two straight lines is empty. In case it is not empty, then a small shift of one of the straight lines can make it empty. The intersection of a plane and a straight line is, as a rule, a point, and that of two planes is a straight line. To describe a reaction polyhedron, the first thing to do is to specify its nodes (vertices). They are the intersections of a plane specified by eqn. (37) with some faces of the set (a cone) of non-negative vectors. Faces of this cone are specified by the sets of equations and unequalities... 

Originally a-acids were estimated gravimetrically as their lead salts (see [22]) but this lengthy procedure has been superceded by conductometric estimation of the end point in the titration of a-acids against lead acetate. Typically [24, 25, 26] a toluene extract of hops is diluted with methanol and the conductivity measured. Small aliquots (0-2 ml) of a methanolic solution of lead acetate (2 or 4 %) are added and the conductivity measured after each addition. A graph is plotted of conductivity against volume of lead acetate solution added and the end point determined from the intersection of two straight lines (Fig. 13.3). The curves may show different shapes in other solvents [27]. 

As mentioned above, the test may also assess the negative effect of water on the bituminous mixture (moisture susceptibility of aggregates). This is achieved by determining the stripping inflection point, which is the point of intersection of two straight lines (see Figure 7.25). 

The rise in reaction rate is due to the expenditure of antioxidant. When it is used up, the reaction proceeds at the constant rate of an uninhibited reaction. The time of antioxidant (t) is determined graphically by the intersection of two straight lines on the kinetic curve. One of these is the straight-line portion of the kinetic curve after AO has been used up. The other is the tangent to the kinetic curve whose inclination angle is one- half the tangent angle of the first. 

The total number of acidic groups on HA was determined by a conductometric titration [22]. A typical conductometric titration curve of commercial HA (Aldrich Chemical Co.) with HNO3 is shown in Fig. 2. The intersection of two straight lines defines the end point of the titration. The total number of acidic groups was determined from the titration curve to be 3.40 x lO mol/g, Since the HA used here was in the form of sodium salt, no end point was obtained from the titration with NaOH. 

Alkanes may also form ring systems to make compounds called cycloalkanes. The naming of these compounds is very similar to the branched kanes, except the cyclo- prefix is used on the parent name. In the condensed structurad formula, the ring is often drawn as lines where the intersection of two straight lines indicates a carbon atom, and the hydrogen atoms aren t shown at all. Figure 14-6 shows both the expanded and condensed form of 1,3-dimethyl cyclohexane. 

Hecht et al. [166] found that the surfactants bind cooperatively on the block copolymer molecules, and then the hydrophobic block changes to hydrophilic. At very low SDS concentrations, the micellization peak of the copolymer was not influenced hence, no binding occurred, or this binding of SDS to the poloxamer was reversible. From the plot of the micellization enthalpy versus SDS concentration, it is possible to determine the concentration at which SDS starts to adsorb on the poloxamer as the intersection of two straight lines (Fig. 19 [166]). From the difference between this concentration and that at which becomes zero, the amount of bound SDS can be estimated. 

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