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The Cartesian y value is y = r sin f sin q = 31/2 sin (p /4) sin (3p /4) = 31/2 21/2/2 21/2/2 = 31/2/2 The Cartesian z value is z = r cos f = 31/2 cos (p /4) = 31/2 21/2/2 = 61/2/2 Our Cartesian ordered triple is therefore P = (x,y,z) = ( 31/2/2,31/2/2,61/2/2) Now let s convert these coordinates to their cylindrical counterparts We have x = 31/2/2 and y = 31/2/2 To find the cylindrical direction angle q, we use the formula q = p + Arctan ( y /x) because x < 0 and y > 0 When we plug in the values for x and y, we get q = p + Arctan [(31/2/2) / ( 31/2/2)] = p + Arctan ( 1) = p + ( p /4) = 3p /4 This is the same as the horizontal direction angle in the original set of spherical coordinates, as we should expect (If things hadn t come out that way, we d have made a mistake!) When we input the values for x and y to the formula for the cylindrical radius r, we get r = [( 31/2/2)2 + (31/2/2)2]1/2 = (3/4 + 3/4)1/2 = (6/4)1/2 = 61/2/2 We calculated that z = 61/2/2, so the cylindrical height h is h = z = 61/2/2 We ve found that the cylindrical equivalent point is (q,r,h) = (3p /4,61/2/2,61/2/2)

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A network manager for a construction company was responsible for supporting 200 users and performing all projects to grow the network aggressively This construction company was committed to developing and using the best technology for construction estimation, engineering, and project management, and as a result had cutting-edge IT systems It was one of the rst medium-sized construction companies to consider and implement wireless technology The network manager, Glenn, felt he had too much work to do His boss, the COO, thought that Glenn was not very e ective Glenn decided to use the method in this book to organize his work He identi ed each project on which he was working, and, with a bit of help, wrote a quick project overview for each one He was working on 42 projects all by himself, as well as keeping everything running! He built a spreadsheet as a master list of projects.

This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Describe the graphs of the following equations in cylindrical coordinates What would they look like in Cartesian xyz space q=0 r=0 h=0 2 Plot the point (q,r,h) = (3p /4,6,8) in the cylindrical coordinate system 3 Consider the point (q,r,h) = (p /4,0,1) in cylindrical coordinates Find the equivalent of this point in Cartesian xyz space 4 Consider the point ( 4,1,0) in xyz space Find the equivalent of this point in cylindrical three-space First, find the exact coordinates Then, using a calculator, approximate the irrational coordinates to four decimal places 5 In the chapter text, we used the conversion formulas to find that the cylindrical equivalent of (x,y,z) = (1,1,1) is (q,r,h) = (p /4,21/2,1) Convert these coordinates back to Cartesian xyz coordinates to verify that the result we got was correct and unambiguous 6 Describe the graphs of the following equations in spherical coordinates What would they look like in Cartesian xyz space q=0 f=0 r=0 7 Plot the point (q,f,r) = (3p /4,p /4,8) in the spherical coordinate system 8 Consider the point (q,f,r) = (p /4,0,1) in spherical coordinates Find the equivalent of this point in Cartesian xyz space 9 Consider the point ( 4,1,0) in xyz space Find the equivalent of this point in spherical three-space First, find the exact coordinates Then, using a calculator, approximate the irrational coordinates to four decimal places 10 Work the final challenge backward to verify that we did our calculations correctly Consider the point P in cylindrical three-space given by P = (q,r,h) = [3p /4,61/2/2,61/2/2] Find the coordinates of P in Cartesian coordinates, and from there, convert to spherical coordinates.

This differential equation is descriptive of a second-order undamped system. The U tube resonates at a natural period established by the