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Extensions of Algebra and Geometry 1) Is it possible for a positive number to exceed its reciprocal by exactly 1? One number that comes close is 8/5, because 8/5 – 5/8 is 39/40. Is there a fraction that comes closer? 2) Find the number that is two thirds of the way (a) from −7 to 17; (b) from m to n. a c is not equivalent to b d ac a that is in fact between the numbers and bd b 3) It is well known that ac . Suppose that a, b, c, and d are all positive. Show bd c a c , while is not. d b d 4) Imagine covering an unlimited plane surface with a single layer of pennies, arranged so that each penny touches six others tangentially. What percentage of the plane is covered? 5) The sides of a right triangle are x − y, x, and x + y, where x and y are positive numbers, and y < x. Find the ratio of x to y. M at hem at ics 1 6) What is the exact value of the expression x2 – 5 when x = 2+√5? Sam’s Warehouse, a member pays $25 a year for membership, and buys at t he t ore prices. A non-member does not pay t he membership fee, but does pay an 7) Using a driver on the 8th tee, which is on a plateau 10 yards above the level fairway, Dale hits another al 5% above t he store prices. Under what condit ions would it make sense t o buy fine shot. Explain why the quadratic function y = 10.0 + 0.5x − 0.002x 2 describes this parabolic trajectory, ership? shown in the figure below. Why should you expect this tee shot to go more than 250 yards? Estimate the length calculator to 1find more accurate 2 ch t he graphs of yof = this x 2 +shot, 5, ythen = xuse − your 4, and y = x2 + onat he same axes. value. WhatHow does this trajectory relate to the trajectory for the drive on the previous hole? ect of the value of c in equat ions of t he form y = x 2 + c? y ........................................................ math I I . Again using a driver on t he 8t h .................... ............. ............. .......... .......•... ......... . . . . . . . . ........ h is on a plateau 10 yards above t he level .................... ....... ....... ...... ...................... ...... .. . . . . . .... ...... Dale hit s anot her fine shot . Explain why .......... .... x 2 rat ic funct ion y = 10.0+ 0.5x − 0.002x describes t his parabolic trajectory, shown findexpect the length theshot shott without a calculator, must set y equal to 0 and solve for gure above. (Continuation) Why should To you this of tee o go more t han 250you yards? 2 why, how to arrive or at xt o−find 250xa=more 5000.accurat e value. e t he lengt hx.ofExplain t his shot , tand henshow use your calculat (a) The next step in the solution process is to add to both hole? sides of this equation. s t his t raject ory relat e t o t he t rajectory for the drive on the previous (b) Complete the solution by showing that the length of the shot is . How does this t inuat ion) To find twhich he lengt h of 268.6, t he shot without calculator, must set y number, is about compare witha your previousyou calculation? 2 0 and solve(c) forComment x. Explain why, and show how t o arrive at x − 250x = 5000.is its significance? on the presence of the number 125 in the answer. What 2 next st ep in t he solut ion process is t o add 125 t o bot h sides of t his equat ion. s t his number chosen? 8) The graph of a quadratic function intersects the x-axis at 0√and at 8. Draw two parabolas that fit this plet e t he solut ion by showing that the length of tHow he shot 125 + 20625. How description and find equations for them. manyisexamples are possible? number, which is about 268.6, compare with your previous calculat ion? ment on the9)presence t he number 125 in t he answer.axis What is it s significance? There is aofunique parabola whose symmetry is parallel to the y-axis, and that passes through the three points (1, 1), (−2, −2), and (0, −4). Write an equation for it. 2 2 2 ph t he equat10) ions on tany he same syst emmust of axes: , y = 0.5x , y pass = 2xthrough , and them? Explain. Given three points, thereybe= a xparabola that will 2 What is t he effect of a in equat ions of t he form y = ax ? t he point s A = (4, 0), B = (4, 5), C = (0, 7), and D = (0, 0). ... ........... .................. ......... ........... .. ........... ......... .. ................. ... . ... . ......... .. 11) What number is exactly midway between 23 − √17 and 23 + √17? What number is exactly midway b b2 4ac b b2 4ac between and ? 2a 2a 12) Robin is mowing a rectangular field that measures 24 yards by 32 yards, by pushing the mower around and around the outside of the plot. This creates a widening border that surrounds the un-mowed grass in the center. During a brief rest, Robin wonders whether the job is half done yet. How wide is the uniform mowed border when Robin is half done? 13) A stop sign — a regular octagon — can be formed from a 12-inch square sheet of metal by making four straight cuts that snip off the corners. How long, to the nearest 0.01 inch, are the sides of the resulting polygon? 16) One stick is twice as long as another. You break the longer stick at a random point. Now you have three sticks. What is the probability that they form a triangle? 17) One stick is three times as long as another. You break the longer stick at a random point. Now you have three sticks. What is the probability that they form a triangle? 18) Find the sum 1 + 2 + 3 + ··· + 100. 19) Find the sum of A1+ A3 + A2 + ··· + An 20) Without using a calculator, find the value of when and .