Mathematics 3{4 Mathematics Department Phillips Exeter Academy Exeter, NH July 2020 To the Student Contents: Members of the PEA Mathematics Department have written the material in this book. As you work through it, you will discover that algebra, geometry, and trigonometry have been integrated into a mathematical whole. There is no Chapter 5, nor is there a section on tangents to circles. The curriculum is problem-centered, rather than topic-centered. Techniques and theorems will become apparent as you work through the problems, and you will need to keep appropriate notes for your records | there are no boxes containing important theorems. There is no index as such, but the reference section that starts on page 101 should help you recall the meanings of key words that are defined in the problems (where they 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 inserted into the problem texts. It is important to make accurate diagrams. Here are a few useful strategies to keep in mind: create an easier problem, use the guess-and-check technique as a starting point, work backwards, recall work on a similar problem. It is important that you work on each problem when assigned, since the questions you may have about a problem will likely motivate class discussion the next day. Problem solving requires persistence as much as it requires ingenuity. When you get stuck, or solve a problem incorrectly, back up and start over. Keep in mind that you're probably not the only one who is stuck, and that may even include your teacher. If you have taken the time to think about a problem, you should bring to class a written record of your efforts, not just a blank space in your notebook. The methods 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 communicate ideas are just as important as getting the correct answer. Technology: Many of the problems in this book require the use of technology (graphing calculators, computer software, or tablet applications) in order to solve them. You are encouraged to use technology to explore, and to formulate and test conjectures. Keep the following guidelines in mind: write before you calculate, so that you will have a clear record of what you have done; be wary of rounding mid-calculation; pay attention to the degree of accuracy requested; and be prepared to explain your method to your classmates. If you don't know how to perform a needed action, there are many resources available online. Also, if you are asked to \graph y = (2x−3)=(x+1)", for instance, the expectation is that, although you might use a graphing tool to generate a picture of the curve, you should sketch that picture in your notebook or on the board, with correctly scaled axes. Standardized testing: Standardized tests like the SAT, ACT, and Advanced Placement tests require calculators for certain problems, but do not allow devices with typewriter-like keyboards or internet access. For this reason, though the PEA Mathematics Department promotes the use of a variety of tools, it is still essential that students know how to use a hand-held graphing calculator to perform certain tasks. Among others, these tasks include: graphing, finding minima and maxima, creating scatter plots, regression analysis, and general numerical calculations. Mathematics 3{4 1. From the top of Mt Washington, which is 6288 feet above sea level, how far is it to the horizon? Assume that the earth has a 3960-mile radius (one mile is 5280 feet), and give your answer to the nearest mile. 2. In mathematical discussion, a right prism is defined to be a solid figure that has two parallel, congruent polygonal bases, and rectangular lateral faces. How would you find the volume of such a figure? Explain your method. 3. A chocolate company has a new candy bar in the shape of a prism whose base is a 1-inch equilateral triangle and whose sides are rectangles that measure 1 inch by 2 inches. These 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 that measures 8 cm by 12 cm by 12 cm? How many of these bars can be packed in rectangular box that measures 8 cm by 5 cm by 5 cm? How would you pack them? 5. Starting at the same spot on a circular track that is 80 meters in diameter, Hillary and Eugene 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 when they finish? There is more than one way to interpret the word distance in this question. 6. Choose a positive number θ (Greek \theta") less than 90.0 and use a calculator to find sin θ and cos θ. Square these numbers and add them. Could you have predicted the sum? .................. ............................................................ .. ............ ....................................... ............ ....... .. ............ ... .... ............................................. 7. Playing cards measure 2.25 inches by 3.5 inches. A full deck ... .... .............................................. .. ... .... .............................................. ................. ... ... ........ ................. .... ... .................. ... ... ... ... ... .. of fifty-two cards is 0.75 inches high. What is the volume of a .. ... ........ ... ... ................................................ .. ... .............. ................................. ...... ... ................ ................................. ....... ................... ................................. ....... ... ......... ... ....... deck of cards? If the cards were uniformly shifted (turning the ... ............ .. ....... ... ........... ............................. ... ............ .......................... .. ......... .......................... .. .. .. ........................... ... .. .. ........................... .. bottom illustration into the top illustration), would this volume .. .......... ....... .. ...................... ... ...................... ... ...................... ... ...................... ... ...................... be affected? ........................ 8. In the middle of the nineteenth century, octagonal barns and silos (and even some houses) became popular. How many cubic feet of grain would an octagonal silo hold if it were 12 feet tall and had a regular base with 10-foot edges? 9. Build a sugar-cube pyramid as follows: First make a 5 × 5 × 1 bottom layer. Then center a 4 × 4 × 1 layer on the first layer, center a 3 × 3 × 1 layer on the second layer, and center a 2 × 2 × 1 layer on the third layer. The fifth layer is a single 1 × 1 × 1 cube. Express the volume of this pyramid as a percentage of the volume of a 5 × 5 × 5 cube. 10. (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, express the 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? July 2020 1 Phillips Exeter Academy Mathematics 3{4 11. A vector v of length 6 makes a 150-degree angle with the vector [1; 0], when they are placed tail-to-tail. Find the components of v. 12. Why might an Earthling believe that the sun and the moon are the same size? 13. Given that ABCDEF GH is a cube (shown at right), what is significant about the square pyramids ADHEG, ABCDG, ........E and ABF EG? ............ .......... ............ ......... ............ ......... ............. ......... ............. ......... ....... ......... F. ....... ......... ....... .......... H . ....... .......... ....... .......... 14. To the nearest tenth of a degree, find the size of the angle . ....... .......... ....... .......... ....... ........... ...G........... formed by placing the vectors [4; 0] and [−6; 5] tail-to-tail at the . origin. It is understood in questions such as this that the answer . .....A. .... ... ... .. .... ... .... ... ... .. is smaller than 180 degrees. ..... ... ...... ... ...... .... B ...... ........ ...... ........ ...... ........ D ...... ........ ...... ........ ...... ........ ...... ........ 15. Flying at an altitude of 39 000 feet one clear day, Cameron ...... ........ ............... looked out the window of the airplane and wondered how far it C was to the horizon. Rounding your answer to the nearest mile, answer Cameron's question. ............ .................... .... .................... .... 16. A triangular prism of cheese is measured and found to be 3 inches .................... .... .................... .... .................... .... ................. ................. ................. ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ................... ....... .................. .... ....... .................. .... tall. The edges of its base are 9, 9, and 4 inches long. Several con- . ....... .................. ... ....... .................