2009 AMC 12 B Problem 7 Represent the price at the end of March with the price

# 2009 amc 12 b problem 7 represent the price at the

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2009 AMC 12 B, Problem #7 “Represent the price at the end of March with the price at the beginning of January.” Solution Answer (B): Let p denote the price at the beginning of January. The price at the end of March was (1 . 2)(0 . 8)(1 . 25) p = 1 . 2 p . Because the price at the end of April was p , the price decreased by 0 . 2 p during April, and the percent decrease was x = 100 · 0 . 2 p 1 . 2 p = 100 6 16 . 7 . To the nearest integer, x is 17. Difficulty: Medium NCTM Standard: Algebra Standard: use symbolic algebra to represent and explain mathematical relationships. Mathworld.com Classification: Number Theory > Arithmetic > Fractions > Percent Segment BD and AE intersect at C , as shown, AB = BC = CD = CE , and A = 5 2 B . What is the degree measure of D ? A B C D E (A) 52 . 5 (B) 55 (C) 57 . 5 (D) 60 (E) 62 . 5 2009 AMC 10 B, Problem #9 “Notice that 4 ABC and 4 CDE are isosceles.” Solution Answer (A): Because 4 ABC is isosceles, A = C . Because A = 5 2 B , we have 5 2 B + 5 2 B + B = 180 , so B = 30 . Therefore ACB = DCE = 75 . Because 4 CDE is isosceles, 2 D + 75 = 180 , so D = 52 . 5 . Difficulty: MediumNCTM Standard:Geometry Standard: analyze properties and determine attributes of two- andthree-dimensional objects.Mathworld.com Classification:Geometry>Plane Geometry>Triangles>Special Triangles>Other Triangles>Isosceles Triangle A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point x meters above the ground so that the upper part, still attached to the stump, touches the ground 1 meter away from the base. What is x ? (A)2.0(B)2.1(C)2.2(D)2.3(E)2.4 2009 AMC 10 B, Problem #10 “Draw a figure to represent the situation.” Solution Answer (E): Let x be the height of the stump. Then 5 - x is the height of the snapped part, now forming the hypotenuse of a right triangle. By the Pythagorean Theorem, x 2 + 1 2 = (5 - x ) 2 = x 2 - 10 x + 25 from which x = 2 . 4 . x 5- x 1 Difficulty: Medium NCTM Standard: Geometry Standard: analyze properties and determine attributes of two- and three-dimensional objects. Mathworld.com Classification: Geometry > Plane Geometry > Triangles > Special Triangles > Other Triangles > Right Triangle How many 7 digit palindromes (numbers that read the same backward as forward) can be formed using the digits 2, 2, 3, 3, 5, 5, 5 ? 2009 AMC 10 B, Problem #11 “Because the digit 5 appears three times, 5 must bethe middle digit of any such palindrome.” Solution Answer (A): Because the digit 5 appears three times, 5 must be the middle digit of any such palindrome. In the first three digits each of 2, 3, and 5 must appear once and the order in which they appear determines the last three digits. Since there are 3! = 6 ways to order three distinct digits the number of palindromes is 6. Difficulty: Medium NCTM Standard: Number and Operations Standard: compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions. Mathworld.com Classification: Number Theory > Special Numbers > Palindromic Numbers > Palindromic Number Distinct points A , B , C , and D lie on a line, with AB = BC = CD = 1 . Points E  #### You've reached the end of your free preview.

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