Visualizing the problem

Visualize the problem. Draw a picture that shows the setup and the information provided in the problem. A picture often helps to interpret and summarize a problem. In this case, the figure should show the spoon, the silver wire, and some indication that atoms are transferred from the wire to the spoon. 

This is a quantitative calculation, so it is appropriate to use the seven-step problem-solving strategy. We are asked to determine an equilibrium constant from standard reduction potentials. Visualizing the problem involves breaking the redox reaction into its two half-reactions ... 

The optimization might well involve more than two variables in a multivariable optimization, but in this case, it is difficult to visualize the problem. 

As soon as I began to visualize the problem, one pos sibility immediately occurred. A few problems with that concept occurred, which seemed to solve themselves rather quickly.. .. Visualizing the required cross section was instantaneous."... 

In studying aerosols it is important to develop in one s mind s eye a picture of the process taking place. By visualizing the problem (even if it is in a simplified form) it is easier to find a method of solution, since most problems are more difficult to set up than they are to solve, once stated. To carry out this visualization, one must have an understanding of the physical phenomena that come into play and a means for estimating their effect. Thus when one is considering a pitched base-... 

To visualize the problem in simple descriptive terms, let us consider the situation for a rider of rectangular cross-section who se area is A = 2l 2b as shown in Fig. 15-1. Let the rate at which heat is generated within the area A be ij units per unit area per unit time. In the moving source problem the average temperature of A will be determined by the rate at which heat flows into the semi-inf inite counterbody. As the source moves in the positive x-direction it encounters material in the track that has been warmed by the heat conducted there from locations of earlier contact. Eventually a state of equilibrium is reached in which the rate that heat is generated in the area of the source is balanced by the rate that it is lost by conduction. The velocity with which the source moves interacts with the rate at which heat flows out of it to influence the temperature distribution within the area of the source, since the rate at which heat is lost by conduction in both senses of the X- and y-axes from the sides of the source area and in the z-direction into the counterbody are also governed by the velocity of translation. 

A very convenient diagram for visualizing the problem of optimizing a multibed adiabatic reactor is the conversion versus temperature plot already encountered in Sec. 11.3, and drawn in Fig. 11.5.d-l for the SO oxidation based on the rate equation of Collina, Corbetta, and Cappelli [113] with an effectiveness factor of 1. (For further reading on this subject see [114] and [115].) This equation is... 

Although we firmly believe that the CPM technique is an invaluable distillation synthesis tool, we would like to point out a shortcoming of the method, namely the ability to deal with higher order systems. The vast majority of problems discussed in this book deal with ternary systems. Ternary systems allow one to visualize the problem graphically and many conclusions can be drawn from this analysis. However, when more components are introduced, the problem becomes exceedingly difficult and insights are diminished. Although a four-component system can be... 

To visualize the problem, consider the case vhth ny = 3. A sphere or an ellipsoid can be looked upon as a very large set of tangent planes (inequality constraints). The Simplex method needs a very large number of infinitesimal movements to achieve the solution and this is an example of an explosion in the number of vertices (called the tile effect) that needs to be analyzed in a three-dimensional space problem ... 

Analyze the problem. Read the problem carefully. What does it 3. say Draw a picture or diagram that will help you to visualize the problem. Write down both the data you are given and the quantity you need to obtain (the unknown). 

Denote the best response function of the retailer by Rr Q) and the best response function of the wholesaler by Ryj A) both defined for zero initial inventory. At this point we have not demonstrated uniqueness or even existence of the equilibrium. It helps, however, to visualize the problem first. We begin by presenting the game graphically (see Figure 14.1, parameters are taken from the example in Section 6 with r = 8). The point A, is a Nash... 

The second step is to develop a plan for solving the problem. The plan should show how the information given is to be used to find the imknown. In the process, reread the problem to make sure you have gathered all the necessary information. It is often helpful to draw a picture that represents the problem. For example, if you were asked to determine the volume of a crystal given its dimensions, you could draw a representation of the crystal and label the dimensions. This drawing would help you visualize the problem. 

Students in the experimental group also indicated that the software helped them to visualize the problem, whieh made the problem more comprehensible to them. 

---