Simplifying formulas

The total charge was based on the formula C = (rate xrri) + ( 30 x d). This formula says that the charge, C, is equal to the mileage rate times the number of miles, m, plus 30 times the number of days, d. Clay has all of the information he needs except for the rate per mile. Filling in the information that he has, Clay knows that 310 = (rate x 400 miles) + ( 30 x 3 days). Letting the mileage rate be represented by / , the formula simplifies to 310 = 400/ + 90. Subtract 90 from each side to get that 220 = 400/ . Divide each side by 400, and you get that R = 0.55. So it appears that the mileage rate was 55[Pg.134]

Let us recapitulate what we have said about compound nuclear reactions. We have said that they are basically nuclear reactions with a long-lived reaction intermediate, which is formed by a complicated set of two-body interactions. We can write down a set of equations that describes the overall compound nuclear cross section. We have shown how this general formula simplifies for specific... 

For identical lattice structure the above formulae simplify significantly since the reciprocal lattice vectors and the areas of the unit cells are the same ... 

Analytical expressions forA(/) of circular cylinders are known. The formulas simplify for infinitely long cylinders [4]. Indeed, the first moment 7 coincides with the diameter d, which is important for the following. The transition from a limited right circular cylinder to an unlimited one is studied in Fig. 1. Here, d-l is a surprisingly exact approximation, if the cylinder is longer than 2d. 

If the radiative widths Tg and Hy are very small and the background constant Kay is neglected, then the formula simplifies to the following useful expression ... 

This formula simplifies considerably when all molecules are initially in the ground state and when det is a continuant, as it b when only transitions between nearest neighbor states occur (for example in the case of the simple harmonic oscillator model). We recall that A as defined in Eq. IV.6 has the property... 

If the distribution pi(ri) is likewise reduced, the formula simplifies to the Coulomb potential energy between two point charges... 

If there is no Poisson ratio mismatch between the metal and oxide, the formula simplifies to Equation (5.35) ... 

Indifferent Substances The chemical potential of homogeneous mixtures can be applied so that reactions between mixed phases can be treated exactly like reactions between pure substances. As an example the chemical drive Amix for the mixing process of two substances that are indifferent to each other should be determined. Because the conversion numbers va and vb coincide with the mole fractions Xa and Xb, the conversion formula simplifies to... 

Fig. 9. Glutathione, ophthalmic Glutathione, lower formula simplified line drawing acid Ophthalmic acid CH3 instead of SH...

Fig. 9.30 Comparison of the compression creep property of nanocomposites with those of existing silicon-nitride ceramics (additive in weight percentage unless specified, molecular formula simplified for clarity. For instance, 6YO" in figure legend stands for 6 wt%Y203 ) [38]. With kind permission of John Wiley and Sons...

In water at 25° C, the last formula simplifies to k = 0.304/ nm when / is expressed in mole per liter (M). When I =1.0 M, for example, c is about 0.3 nm, and when I = 0.001 M, k is about 10 nm. The largest experimentally obtainable values of K are in the range of tens to hundreds of nanometers, values characteristic of dilute, essentially salt-free polyelectrolyte solutions. For these extreme cases, I is dominated by the bulk concentration of counterions dissociated as the polymer dissolves. How this concentration can be estimated is discussed later for highly charged polyelectrolytes, not all dissociated counterions contribute to I. 

When the sphere is small compared to the size of the tube, the formula simplifies to... 

This is a simplified formula for tartar emetic, for X-ray crystal analysis and infrared studies indicate that the. Sb is a part of the antimonate anion [Sb(OH)4] and forms part of a cyclic system. 

SMILES (simplified molecular-input line-entry specification) a way of specifying a molecular formula and connectivity, but not the three-dimensional geometry... 

With practice writing structural formulas for organic molecules soon becomes rou tine and can be simplified even more For example a chain of carbon atoms can be rep resented by drawing all of the C—C bonds while omitting individual carbons The result mg structural drawings can be simplified still more by stripping away the hydrogens... 

In these simplified representations called bond line formulas or carbon skeleton dia grams, the only atoms specifically written m are those that are neither carbon nor hydro gen bound to carbon Hydrogens bound to these heteroatoms are shown however... 

To simplify the formulae in this subsection we shall write w, W, w,. .. instead of w t),. This means that we fix t and consider... 

Let us now obtain a complete system of boundary conditions fulfilled at Lc provided that the simplified nonpenetration condition (3.185) holds. We assume the solution x G iX is smooth enough and use Green s formulas for smooth functions (see Section 1.4),... 

To simplify the formulae below we assume that Cijki = SlSf, 5 is the Kronecker symbol. Nevertheless, all the results obtained in the section are valid in the general case (5.74). 

The dependence of solutions to (5.79)-(5.82) on the parameters a, 5 is not indicated at this step in order to simplify the formulae. Note that boundary conditions (5.81) do not coincide with (5.71) the conditions (5.81) can be viewed as a regularization of (5.71) connected with the proposed regularization of the equilibrium equations (5.68). Also, the artificial initial condition for o is introduced. 

Again, to simplify the formulae we assume Cijki =. Recall that the set... 

Noncomputerized Calculation. Design charts or tables are used for hand calculations or simplified formulas such as the guided cantilever equation given ia Reference 31, described briefly below. 

Substitution of Eqs. (5-262) through (5-265) into Eq. (5-261) results in the following simplified formula ... 

Air and a number of other gases have a value of k = 1.39 to 1.41. To simplify calculations for these gases, tables have been made of the bracketed expression [(p2/pi) 1] these equations for a value of k = 1.395. These are known as X factors, and they are given in Table 10-13. By using X factors, the adiabatic formulas for k = 1.395 read as follows ... 

Based on IEEE-80 recommendations, the following simplified formula may be used to determine the zero sequence current in the event of a phase to ground fault ... 

This is a generalized formula to simplify the equation. In fact, the actual switching current will be much less than this, due to circuit impedance. For more accurate... 

Often in stress analysis we may be required to make simplified assumptions, and as a result, uneertainties or loss of aeeuraey are introdueed (Bury, 1975). The aeeuraey of ealeulation deereases as the eomplexity inereases from the simple ease, but ultimately the eomponent part will still break at its weakest seetion. Theoretieal failure formulae are devised under assumptions of ideal material homogeneity and isotropie behaviour. Homogeneous means that the materials properties are uniform throughout isotropie means that the material properties are independent of orientation or direetion. Only in the simplest of eases ean they furnish us with the eomplete solution of the stress distribution problem. In the majority of eases, engineers have to use approximate solutions and any of the real situations that arise are so eomplieated that they eannot be fully represented by a single mathematieal model (Gordon, 1991). 

This allows for the equivalence between crossed cylinders and the particle on a plane problem. Likewise, the mechanics of two spheres can be described by an equivalently radiused particle-on-a-plane problem. The combination of moduli and the use of an effective radius greatly simplifies the computational representation and allows all the cases to be represented by the same formula. On the other hand, it opens the possibility of factors of two errors if the formula are used without realizing that such combinations have been made. Readers are cautioned to be aware of these issues in the formulae that follow. 

A common process task involves heating a slurry by pumping it through a well-stirred tank. It is useful to know the temperature profile of the slurry in the agitated vessel. This information can be used to optimize the heat transfer process by performing simple sensitivity studies with the formulas presented below. Defining the inlet temperature of the slurry as T, and the temperature of the outer surface of the steam coil as U then by a macroscopic mass and energy balance for the system, a simplified calculation method is developed. 

Simplified formulas are applied in engineering design. For example, in the case of an air curtain with heated indoor air the necessary temperature of the supplied air may be calculated according to the following formula ... 

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