Use your spreadsheet to i. A simulation is an experiment, model, or activity that imitates real or hypothetical conditions. The newspaper article shown here describes how astrophysicists used computers to simulate a collision between Earth and a planet the size of Mars, an event that would be impossible to measure directly.
The simulation showed that such a collision could have caused both the formation of the moon and the rotation of Earth, strengthening an astronomical theory put forward in the s. For each of the following, describe what is being simulated, the advantages of using a simulation, and any drawbacks. In some situations, especially those with many variables, it can be difcult to calculate an exact value for a quantity.
In such cases, simulations often can provide a good estimate. Simulations can also help verify theoretical calculations. Example 1 Simulating a Multiple-Choice Test. When writing a multiple-choice test, you may have wondered What are my chances of passing just by guessing? Suppose that you make random guesses on a test with 20 questions, each having a choice of 5 answers. Intuitively, you would assume that your mark will be somewhere around 4 out of 20 since there is a 1 in 5 chance of guessing right on each question.
However, it is possible that you could get any number of the questions rightanywhere from zero to a perfect score. Devise a simulation for making guesses on the multiple-choice test. Run the simulation times and use the results to estimate the mark you are likely to get, on average.
Would it be practical to run your simulation times or more? Shufe the ve cards and choose one at random. If it is the designated card, then you got the rst question right.
If one of the other four cards is chosen, then you got the question wrong. Put the chosen card back with the others and repeat the process 19 times to simulate answering the rest of the questions on the test. Keep track of the number of right answers you obtained. However, you would have to choose a card times, which would be quite tedious. Instead, form a group with some of your classmates and pool your results, so that each student has to run only 10 to 20 repetitions of the simulation.
Make a table of the scores on the simulated tests and calculate the mean score. You will usually nd that this average is fairly close to the intuitive estimate of a score around 4 out of However, a mean does not tell the whole story. Tally up the number of times each score appears in your table. Now, construct a bar graph showing the frequency for each score. Your graph will look something like the one shown. This graph gives a much more detailed picture of the results you could expect.
Although 4 is the most likely score, there is also a good chance of getting 2, 3, 5, or 6, but the chance of guessing all 20 questions correctly is quite small.
Make sure both lists are empty. This function produces random integers. Enter 1 for the lower limit, 5 for the upper limit, and 20 for the number of trials. L1 will now contain 20 random integers between 1 and 5. Press 2nd 1 to enter L1 into the sort function. When you return to L1, the numbers in it will appear in ascending order. Now, you can easily scroll down the list to determine how many correct answers there were in this simulation.
See Appendix B for more details on how to use the graphing calculator and software functions in Solutions 2 to 4. Again, you may want to pool results with your classmates to reduce the number of times you have to enter the same formula over and over.
If you know how to program your calculator, you can set it to re-enter the formulas for you automatically. However, unless you are experienced in programming the calculator, it will probably be faster for you to just re-key the formulas. Turn off all plots except Plot1. For Type, choose the bar-graph icon and enter L2 for Xlist.
Freq should be 1, the default value. Set Xscl to 1 so that the. However, the maximum list length on the TI Plus is , so you would have to use at least two lists to run the simulation a times or more. The RAND function produces a random real number that is equal to or greater than zero and less than one. The INT function rounds a real number down to the nearest integer.
Combine these functions to generate a random integer between 1 and 5. Record this score in cell A Fill feature. Then, use the average function to nd the mean score for the simulated tests. Record this average in cell B Then, enter 0 in cell B26 and highlight cells B26 through V Use the Fill feature to enter the integers 0 through 20 in cells B26 through V In B27, enter the formula for the number of zero scores; in C27, the number of 1s; in D27, the number of 2s; and so on, nishing with V27 having the number of perfect.
Finally, use the Chart feature to plot frequency versus score. Launch FathomTM and open a new document if necessary. Drag a new collection box to the document and rename it MCTest. Right-click on the box and create 20 new cases. Drag a case table to the work area. You should see your 20 cases listed. Expand the table if you cannot see them all on the screen.
Enter 1,5 into the randomInteger function and click OK to ll the Guess column with random integers between 1 and 5. Scroll down the column to see how many correct guesses there are in this simulation.
FathomTM to automatically repeat the simulation times automatically and keep track of the number of correct guesses. First, set up the count function. Rightclick on the collection box and select Inspect Collection. Click on the MCTest collection box. Click on this box and drag a new case table to the document. FathomTM will automatically run ve simulations of the multiple-choice test and show the results in this case table.
Turn off the animation in order to speed up the simulation. Change the number of measures to Then, click on the Collect More Measures button. You should now have measures in the case table for Measures from MCTest. Next, use the mean function to nd the average score for these simulations.
Select mean, enter Score between the brackets, and select OK to display the mean mark on the tests. Finally, plot a histogram of the scores from the simulations. Drag the graph icon onto the work area. Then, drag the Score column from the Measures from MCTest case table to the horizontal axis of the graph. FathomTM then automatically produces a dot plot of your data. To display a histogram instead, simply click the menu in the upper right hand corner of the graph and choose Histogram.
Key Concepts Simulations can be useful tools for estimating quantities that are difcult to calculate and for verifying theoretical calculations. A variety of simulation methods are available, ranging from simple manual models to advanced technology that makes large-scale simulations feasible. Make a table summarizing the pros and cons of the four simulation methods.
A manufacturer of electric motors has a failure rate of 0. A quality-control inspector needs to know the range of the number of failures likely to occur in a batch of of these motors. Which tool would you use to simulate this situation? Give reasons for your choice. Practise A 1. Write a graphing calculator formula for a generating random integers between.
On the oddnumbered tosses, walk one step north for heads and one step south for tails. On evennumbered tosses, walk one step east for heads and one step west for tails.
Cartesian graph, simulate this random walk for steps. Note the coordinates where you nish. Describe three other manual methods you. Make a histogram of the sums. Communication a Describe a calculation or mechanical. Explain why or why not. Application A brother and sister each tell the. What sums would you expect to be the most frequent and least frequent? Give reasons for your answers. The brother stated that he owned the car he was driving. The sister said he was telling the truth.
Develop a simulation to show whether you should believe them. Outline a simulation procedure you could use to determine this quantity. Graph theory is a branch of mathematics in which graphs or networks are used to solve problems in many elds. Graph theory has many applications, such as setting examination timetables colouring maps modelling chemical compounds designing circuit boards building computer, communication, or transportation networks determining optimal paths In graph theory, a graph is unlike the traditional Cartesian graph used for graphing functions and relations.
A graph also known as a network is a collection of line segments and nodes. Mathematicians usually call the nodes vertices and the line segments edges. Networks can illustrate the relationships among a great variety of objects or sets. This network is an illustration of the subway system in Toronto. In order to show the connections between subway stations, this map is not to scale. In fact, networks are rarely drawn to scale. In each of the following diagrams the lines represent borders between countries.
Countries joined by a line segment are considered neighbours, but countries joining at only a single point are not. Determine the smallest number of colours needed for each map such. Although the above activity is based on maps, it is very mathematical. It is about solving problems involving connectivity.
Each country could be represented as a node or vertex. Each border could be represented by a segment or edge. D, respectively. D, E, and F, respectively. Note that the positions of the vertices are not important, but their interconnections are. Connect A with edges to B, C, and F only. Use the same process to draw the rest of the edges. As components of networks, edges could represent connections such as roads, wires, pipes, or air lanes, while vertices could represent cities, switches, airports, computers, or pumping stations.
The networks could be used to carry vehicles, power, messages, uid, airplanes, and so on. If two vertices are connected by an edge, they are considered to be adjacent. In the network on the right, A and B are adjacent, as are B and C. A and C are not adjacent.
The number of edges that begin or end at a vertex is called the degree of the vertex. In the network, A has degree 1, B has degree 2, and C has degree 3. The loop counts as both an edge beginning at C and an edge ending at C. Any connected sequence of vertices is called a path. If the path begins and ends at the same vertex, the path is called a circuit. A circuit is independent of the starting point. Instead, the circuit depends on the route taken. Example 2 Circuits A B. Since this path begins at B and ends at A, it is not a circuit.
This path begins at B and ends at B, so it is a circuit. Since this path begins at C and ends at A, it is not a circuit. A network is connected if and only if there is at least one path connecting each pair of vertices. A complete network is a network with an edge between every pair of vertices.
In a traceable network all the vertices are connected to at least one other vertex and all the edges can be travelled exactly once in a continuous path. Traceable: All vertices are connected to at least one other vertex, and the path from A to B to C to D to A to C includes all the edges without repeating any of them. The eighteenth-century German town of Koenigsberg now the Russian city of Kaliningrad was situated on two islands and the banks of the Pregel River.
Koenigsberg had seven bridges as shown in the map. People of the town believedbut could not provethat it was impossible to tour the town, crossing each bridge exactly once, regardless of where the tour started or nished. Were they right? Reduce the map to a simple network of vertices and edges. Let vertices A and C represent the mainland, with B and D representing the islands.
Each edge represents a bridge joining two parts of the town. If, for example, you begin at vertex D, you will leave and eventually return but, because D has a degree of 3, you will have to leave again. Conversely, if you begin elsewhere, you will pass through vertex D at some point, entering by one edge and leaving by another.
But, because D has degree 3, you must return in order to trace the third edge and, therefore,. So, your path must either begin or end at vertex D. Because all the vertices are of odd degree, the same argument applies to all the other vertices.
Since you cannot begin or end at more than two vertices, the network is non-traceable. Therefore, it is indeed impossible to traverse all the towns bridges without crossing one twice. Leonhard Euler developed this proof of Example 3 in He laid the foundations for the branch of mathematics now called graph theory.
Among other discoveries, Euler found the following general conditions about the traceability of networks. A network is traceable if it has only vertices of even degree even vertices or exactly two vertices of odd degree odd vertices. If the network has two vertices of odd degree, the tracing path must begin at one vertex of odd degree and end at the other vertex of odd degree. Example 4 Traceability and Degree. For each of the following networks, a list the number of vertices with odd degree and with even degree b i.
Solution i a 3 even vertices ii a 0 even vertices iii a 3 even vertices iv a 1 even vertex. If it is possible for a network to be drawn on a two-dimensional surface so that the edges do not cross anywhere except at vertices, it is planar. Example 5 Planar Networks. Therefore, the network is non-planar. A graphic designer is working on a logo representing the different tourist regions in Ontario.
What is the minimum number of colours required for the design shown on the right to have all adjacent areas coloured differently? Because the logo is two-dimensional, you can redraw it as a planar network as shown on the right. This network diagram can help you see the relationships between the regions. The vertices represent the regions and the edges show which regions are adjacent. Vertices A and E both connect to the three other vertices but not to each other. Therefore, A and E can have the same colour, but it must be different from the colours for B, C, and D.
Vertices B, C, and D all connect to each other, so they require three different colours. Thus, a minimum of four colours is necessary for the logo.
This example is a specic case of a famous problem in graph theory called the four-colour problem. As you probably conjectured in the investigation at the start of this www. This fact had been suspected out more about the four-colour problem. Write a for centuries but was not proven until The short report on the history of the four-colour proof by Wolfgang Haken and Kenneth Appel at problem.
Non-planar maps can require more colours. Example 7 Scheduling. The mathematics department has ve committees. Each of these committees meets once a month. Draw the schedule as a network, with each vertex representing a different committee and each edge representing a potential conict between committees a person on two or more committees.
Analyse the network as if you were colouring a map. The network can be drawn as a planar graph. Therefore, a maximum of four time slots is necessary to colour this graph. Because Committee A D is connected to the four other committees degree 4 , at least two time slots are necessary: one for committee A and at least one for Project Prep all the other committees.
Because each of the other nodes has degree 3, at least one more time slot is necessary. In fact, three time slots are Graph theory provides sufcient since B is not connected to D and C is not connected to E. Key Concepts In graph theory, a graph is also known as a network and is a collection of line segments edges and nodes vertices.
If two vertices are connected by an edge, they are adjacent. The degree of a vertex is equal to the number of edges that begin or end at the vertex.
A path is a connected sequence of vertices. A path is a circuit if it begins and ends at the same vertex. A connected network has at least one path connecting each pair of vertices. A complete network has an edge connecting every pair of vertices. A connected network is traceable if it has only vertices of even degree even vertices or exactly two vertices of odd degree odd vertices.
If the network has two vertices of odd degree, the tracing must begin at one of the odd vertices and end at the other. A network is planar if its edges do not cross anywhere except at the vertices.
The maximum number of colours required to colour any planar map is four. Describe how to convert a map into a network. Use an example to aid in. A network has ve vertices of even degree and three vertices of odd degree.
Using a diagram, show why this graph cannot be traceable. A modern zoo contains natural habitats for its animals. However, many of. Describe how to use graph theory to determine the number of different habitats required.
Koenigsberg map to make it traceable? Provide evidence for your answer. How many. Hint: Consider each subject to be one vertex of a network. A highway inspector wants to travel each. Draw three other maps using similar lines. Investigate the four maps and make a conjecture of how many colours are needed for this type of map. Determine whether it is possible to do so for each map shown below. B, and C, respectively. Water, gas, and electrical utilities are located at positions D, E, and F, respectively.
Determine whether the houses can each be connected to all three utilities without any of the connections crossing. Provide evidence for your decision. Is it necessary to reposition any of the utilities? Investigate other networks and determine the sum of the degrees of their vertices. The four Anderson sisters live near each. The exterior of the house can be treated as one room. Sketch a oor plan based on this network.
None of these paths intersect. Can their brother Warren add paths from his house to each of his sisters houses without crossing any of the existing paths?
In the diagram below, a sheet of paper with. Given that point A is inside the closed gure, determine whether point B is inside or outside. Provide reasons for your answer. For each network below, determine which links need to be backed up. Describe how to back up the links. In the network shown, all link times are in seconds. Thunder Bay 2. During an election campaign, a politician Winnipeg Montral Charlottetown. The diagram shows the cheapest one-way fare for ights between the cities.
Determine the least expensive travel route beginning and ending in Toronto. Hint: You can usually nd the shortest paths by considering the shortest edge at each vertex. Communication a Find a route that passes through each a Can a connected graph of six vertices. No three people at the table know each other.
Show that at least three of the six people seated at the table must be strangers to each other. Hint: Model this situation using a network with six vertices.
Provide evidence to support your answer. Chapter 1: A Letter to God. Chapter 3: Two Stories about Flying. Chapter 4: From the Diary of Anne Frank. Chapter 5: The Hundred Dresses—I. Chapter 7: Glimpses of India. Chapter 8: Mijbil the Otter. Chapter 9: Madam Rides the Bus. Chapter The Sermon at Benares. Chapter The Proposal. Chapter 1: A Triumph of Surgery. Chapter 3: The Midnight Visitor. It will assist you in improving their answer writing and presenting answers skills in board exam.
One few chapters are there which need extra practice. For example, Light: Reflection and Refraction and Electricity contains numerical problems also. Numerical problems require more practice including some extra questions from various books. For example, practicing Introduction to Trigonometry, chapter 8 needs more and more questions to be confident.
Similarly, geometry portion also requires extra questions to practice. There are total 25 chapters in Social Science Class X. There are total 11 stories and 12 poems present in the Class 10 First Flight textbook which are very important in developing the basic language skills of students and knowing different themes of life.
It will enhance memory and retention skills of students to a great size. The postmaster was a kind, generous, helpful, amiable and God-fearing man. He was generous, as he helped Lencho with 70 pesos. What did the young seagull do to attract the attention of his mother? He came slowly up to the brink of the ledge and stood on one leg. He hid the other leg under his wing. He closed one eye and then the other and pretended to be falling asleep.
Thus he tried to attract the attention of his mother. Why does Valli refuse to look out of the window on her way back? Valli refused to look out of the window on her way back because the memory of the dead cow haunted her, dampening her enthusiasm. Previous Post Next Post. Contact Form. LinkList ul li ul'.
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