Mathematics 3-4 - exeter.edu

1 Mathematics 3-4 Mathematics DepartmentPhillips exeter AcademyExeter, NHAugust 2021To the StudentContents: Members of the PEA Mathematics Department have written the material in thisbook. As you work through it, you will discover that algebra, geometry, and trigonometryhave been integrated into a mathematical whole. There is no Chapter 5, nor is there a sectionon tangents to circles. The curriculum is problem-centered, rather than and theorems will become apparent as you work through the problems, andyou will need to keep appropriate notes for your records there are no boxes containingimportant theorems.

2 There is no index as such, but the reference section that starts on page103 should help you recall the meanings of key words that are defined in the problems (wherethey usually appear italicized).Problem solving: Approach each problem as an exploration. Reading each question care-fully is essential, especially since definitions, highlighted in italics, are routinely insertedinto the problem texts. It is important to make accurate diagrams. Here are a few usefulstrategies to keep in mind: create an easier problem, use the guess-and-check technique as astarting point, work backwards, recall work on a similar problem.

3 It is important that youwork on each problem when assigned, since the questions you may have about a problem willlikely motivate class discussion the next day. Problem solving requires persistence as muchas it requires ingenuity. When you get stuck, or solve a problem incorrectly, back up andstart over. Keep in mind that you re probably not the only one who is stuck, and that mayeven include your teacher. If you have taken the time to think about a problem, you shouldbring to class a written record of your efforts, not just a blank space in your notebook. Themethods that you use to solve a problem, the corrections that you make in your approach,the means by which you test the validity of your solutions, and your ability to communicateideas are just as important as getting the correct : Many of the problems in this book require the use of technology (graphingcalculators, computer software, or tablet applications) in order to solve them.

4 You areencouraged to use technology to explore, and to formulate and test conjectures. Keep thefollowing guidelines in mind: write before you calculate, so that you will have a clear recordof what you have done; be wary of rounding mid-calculation; pay attention to the degree ofaccuracy requested; and be prepared to explain your method to your classmates. If you don tknow how to perform a needed action, there are many resources available online. Also, ifyou are asked to graphy= (2x 3)/(x+ 1) , for instance, the expectation is that, althoughyou might use a graphing tool to generate a picture of the curve, you should sketch thatpicture in your notebook or on the board, with correctly scaled testing:Standardized tests like the SAT, ACT, and Advanced Placementtests require calculators for certain problems, but do not allow devices with typewriter-likekeyboards or internet access.

5 For this reason, though the PEA Mathematics Departmentpromotes the use of a variety of tools, it is still essential that students know how to use ahand-held graphing calculator to perform certain tasks. Among others, these tasks include:graphing, finding minima and maxima, creating scatter plots, regression analysis, and generalnumerical the top of Mt Washington, which is 6288 feet above sea level, how far is it to thehorizon? Assume that the earth has a 3960-mile radius (one mile is 5280 feet), and give youranswer to the nearest mathematical discussion, aright prismis defined to be a solid figure that has twoparallel, congruent polygonal bases, and rectangularlateral faces.

6 How would you find thevolume of such a figure? Explain your chocolate company has a new candy bar in the shape of a prism whose base is a1-inch equilateral triangle and whose sides are rectangles that measure 1 inch by 2 prisms will be packed in a box that has a regular hexagonal base with 2-inch edges,and rectangular sides that are 6 inches tall. How many candy bars fit in such a box?4.(Continuation) The same company also markets a rectangular chocolate bar that mea-sures 1 cm by 2 cm by 4 cm. How many of these bars can be packed in a rectangular box thatmeasures 8 cm by 12 cm by 12 cm?

7 How many of these bars can be packed in a rectangularbox that measures 8 cm by 5 cm by 5 cm? How would you pack them? at the same spot on a circular track that is 80 meters in diameter, Hillary andEugene run in opposite directions, at 300 meters per minute and 240 meters per minute,respectively. They run for 50 minutes. What distance separates Hillary and Eugene whenthey finish? There is more than one way to interpret the worddistancein this a positive number (Greek theta ) less than and use a calculator to findsin and cos . Square these numbers and add them. Could you have predicted the sum?

8 Cards measure inches by inches. A full deckof fifty-two cards is inches high. What is the volume of adeck of cards? If the cards were uniformly shifted (turning thebottom illustration into the top illustration), would this volumebe affected? How might you generalize this?.. the middle of the nineteenth century, octagonal barns and silos (and even somehouses) became popular. How many cubic feet of grain would an octagonal silo hold if itwere 12 feet tall and had a regular base with 10-foot edges? a sugar-cube pyramid as follows: First make a 5 5 1 bottom layer.

9 Thencenter a 4 4 1 layer on the first layer, center a 3 3 1 layer on the second layer, andcenter a 2 2 1 layer on the third layer. The fifth layer is a single 1 1 1 cube. Expressthe volume of this pyramid as a percentage of the volume of a 5 5 5 (Continuation) Repeat the sugar-cube construction, starting with a 10 10 1 base,the dimensions of each square decreasing by one unit per layer. Using a calculator, expressthe volume of the pyramid as a percentage of the volume of a 10 10 10 cube. Repeat,using 20 20 1, 50 50 1, and 100 100 1 bases. Do you see the trend?August 20211 phillips exeter AcademyMathematics 3-411.

10 A vectorvof length 6 makes a 150-degree angle with the vector [1,0], when they areplacedtail-to-tail. Find the components Why might an Earthling believe that the sun and the moon are the same size?13. Given thatABCDEFGHis a cube (shown at right), whatis significant about the square pyramidsADHEG,ABCDG,andABFEG?..ABCDEFGH1 4. To the nearest tenth of a degree, find the size of the angleformed by placing the vectors [4,0] and [ 6,5] tail-to-tail at theorigin. It is understood in questions such as this that the answeris smaller than 180 Flying at an altitude of 39 000 feet one clear day, Cameronlooked out the window of the airplane and wondered how far itwas to the horizon.