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# EXAMS GUIDE 31864

CSU - Dominguez hills

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▯ ▯ SA T /ACT Math And Beyond: Problems Book A Standard High School Workbook First Edition Qishen Huang, Ph.D. This book helps you √ Score highly on SAT/ACT Math section, √ Get ready for Calculus course, and √ Win in high school math contests. ISBN-10: 0-9819072-0ISBN-13: 978-0-9819072-0-8 ▯ Copyright 2008 by Qishen Huang. It is unlawful for anyone to incorporate any part of the content into his works without the author’s permission. Questions for the author should be sent to GoodMathBook@Yahoo.com. Limit of Liability/Disclaimer of Warranty: The author makes no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaims any implied warranties of anything for a particular purpose. The author shall not be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. ACT is a registered trademark of ACT, Inc. SAT is a registered trademark of the Col- lege Entrance Examination Board. Both companies were not involved in the creation and marketing of the book. ISBN-10: 0-9819072-0-2 ISBN-13: 978-0-9819072-0-8 10900 Stonecutter Place, Gaithersburg, MD 20878-4805, USA. ▯ Qishen Huang 2 Preface I performed most of groundwork for this workbook while my oldest son was in high school. With the book, I have three relevant goals for him. 1. To get a perfect or near perfect score on the SAT Math section. It is getting harder for high school students to gain acceptance to a decent college these days because of the large number of applicants across the country. Students can read the article Applications to Colleges Are Breaking Records by Karen W. Arenson dated January 17, 2008 in The New York Times. A top math score gives students an edge against competition. 2. To build a solid foundation for college level math. With the growing number of problems that need to be solved by advancements in science and technology, current and future generations cannot afford to have a weak foundation in mathematics. Johns Hopkins University Mathematics Professor W. Stephen Wilson gave his 2006 calculus class the same test his 1989 class had taken, and the 2006 students were wiped out by the old class. A stitch in time saves nine. 3. To be competitive internationally in math. According to the article U.S. Leaders Fret Over Students’ Math and Science Weak- nesses by Vaishali Honawar of Education Week, US high school students had lower math score than any other developed country. Rep. Vernon Ehlers of Michigan de- clares it a steadily worsening crisis. Central to the crisis is a popular culture that doesn’t value math and science. This problems book will be updated from time to time. A detailed Solutions Manual is available. Relevant inquires should be sent to GoodMathBook@Yahoo.com.. ▯ Qishen Huang 3 Contents 1 Tips on Math Homework 1 2 Algebraic Expressions: Basic 3 3 Algebraic Expressions: Intermediate 7 4 Rational Expressions 9 5 Linear Relations: Basic 13 6 Linear Relations: Intermediate 18 7 Linear Relations: Advanced 21 8 Word Problems: Basic 23 9 Word Problems: Intermediate 25 10 Word Problems: Advanced 27 11 Geometry: Basic 29 12 Geometry: Intermediate 34 13 Geometry: Advanced 39 14 Radicals 44 15 Exponentials: Basic 47 16 Exponentials: Intermediate 52 17 Exponentials: Advanced 54 18 General Functions 57 ▯ Qishen Huang 4 CONTENTS CONTENTS 19 Inverse Functions 60 20 Quadratic Functions: Basic 62 21 Quadratic Functions: Intermediate 66 22 Quadratic Functions: Advanced 68 23 Polynomial and Rational Functions 70 24 Radical Equations and Functions 74 25 Circles 76 26 Ellipses 79 27 Hyperbolas 82 28 Sequences: Basic 86 29 Sequences: Intermediate 91 30 Sequences: Advanced 93 31 Trigonometry: Basic 94 32 Trigonometry: Intermediate 103 33 Trigonometry: Advanced 106 34 Complex Numbers 109 35 Vectors and Matrices 111 36 Parameterized Equations 114 37 Polar Coordinates 116 38 Statistics 118 39 Limits 122 ▯ Qishen Huang 5 Chapter 1 Tips on Math Homework 1.1 To solve any math problem, follow these four steps. (a) Understand the problem. (b) Devise a plan. (c) Carry out the plan. (d) Look back and check. 1.2 Follow the rules of acceptable mathematical writing. (a) Describe your approach at the beginning, if the solution is neither short nor simple. (b) Define variables unless no remote possibility of confusion. (c) Use mathematical notations correctly. (d) Treat mathematical expressions as nouns or sentences. (e) Follow the rules of grammar when combining words and expressions. (f) Make sure that your solution has a single flow. (g) State clearly your result in the final sentence. 1.3 Given a + b = 1, find the value of 2a + 2b. Two solutions are presented below. Only one is correct, even though both yield the correct answer. I. Correct Solution Because a + b = 1, 2a + 2b = 2(a + b) = 2 × 1 = 2. ▯ Qishen Huang 1 CHAPTER 1. TIPS ON MATH HOMEWORK II. Incorrect Solution Because a + b = 1, assume a = 0.5 and b = 0.5. Then 2a + 2b = 2 × 0.5 + 2 × 0.5 = 2. ▯ Qishen Huang 2 Chapter 2 Algebraic Expressions: Basic 2.1 Review basic formulas. (A) (a + b) = a + 2ab + b 2 (B) (a + b + c) = a + b + c + 2ab + 2ac + 2bc (C) (a − b) = a − 2ab + b 2 (D) (a + b) = a + 3a b + 3ab + b 3 (E) (a − b) = a − 3a b + 3ab − b 3 2 2 (F) a − b = (a + b)(a − b) 3 3 2 2 (G) a − b = (a − b)(a + ab + b ) 3 3 2 2 (H) a + b = (a + b)(a − ab + b ) 1 2 1 (I) a + = a + + 2, where a ▯= 0. a a 2 2.2 The middle number of three increasing consecutive odd numbers is n. Express the product of the three numbers in terms of n. 2.3 Four sides of a square are a units long. Express the area and perimeter of the square in terms of a. 2.4 The product of two numbers is 10. One of them is a. Express their sum in terms of a. 2.5 The first of three increasing consecutive even numbers is 2n − 4. Express the last number in terms of n. 2.6 John drives from point A to B at speed of x miles per hour. On his way back, his speed is 10% faster. Which expression is his speed back? (a) x + 0.10 ▯ Qishen Huang 3 CHAPTER 2. ALGEBRAIC EXPRESSIONS: BASIC (b) x × 0.10 (c) x × 1.10 2.7 Sort the following values in ascending order: 2 1 0.5, −0.5, 0.5 , and 0.5. 2.8 Value a satisfies −1 < a < 0. Sort these values in ascending order: a, −a, a 2, and 1 . a 2.9 Find the values of x such that x 2 = 0.36. √ 2 2 2.10 Simplify expression a b , where a < 0 < b. 2.11 Consolidate expression 10x 2+ [2x − (5 + 4x − x) − 3]. 3 2 2 3 2.12 Consolidate expression a − 4 − (a − 5a) + (5a − 3 − 6a ). 2.13 Consolidate expression (x + 1) + x(x − 2y) − 2x. 2 2 2 2 2.14 Simplify expression 3ab − 5a b + (−3a b) − 4ab . 2.15 Suppose a − b + c − d = a − x. Express x in terms of a, b, c, and d. 2.16 Suppose a × x × a m+5 = a2m+8 . Express x in terms of a and m. 2.17 Given equation (x − 1)(x + 7) = (x + 1)(x − 7) + y, express y in terms of x. 3 2 2 2 3 3 2.18 Suppose z ÷ x y = 2x y . Express z in terms of x and y. 2 2.19 Classify the following identities as true or false. (A) (−a 2) = a 6 (B) a + a = a 5 (C) a × a = a 6 3 2 6 (D) (a ) = a 3 3 (E) (3a) = 9a 6 3 2 (F) a ÷ a = a 2.20 Classify the following identities as true or false. (a) (x + 1)2 = 1 + 2x + x 2 ▯ Qishen Huang 4 CHAPTER 2. ALGEBRAIC EXPRESSIONS: BASIC 2 2 (b) (x − 1) = x − 1 3 3 6 (c) x + x = x 2.21 Classify the following identities as true or false. (a) (2xy) ∙ (−3xy) = −6xy (b) (x − y)(x + 2y) = x 2+ xy − 2y 2 (c) (−4x ) = −12x 6 2 2 (d) (x − y) = (y − x) 2.22 Identify the expressions that are always positive. 2 (A) a (B) a + 2 (C) |a + 1| (D) a 2 + 1 (E) 4 − (−a) 3 2.23 Find the expressions that can have value of 0. (a) |x − 1| 2 (b) x + |y| 2 (c) x + |x − 1| 2.24 Does a 4> 0 imply a > 0? 2.25 Non-negative values a and b satisfy a + b = 0. Find the values of a and b. 2.26 Assume |x| = 3, |y| = 10, and xy < 0. Find all possible values of x − y. 2.27 Factor expressions in x. 2 (a) x − 4 4 2 (b) x − 64x 3 2 (c) x − 2x − 4x − 12 (d) (x − 1)(x − 2) − 6 2 2 2 (e) (x − 2x) − 2(x − 2x) − 3 2.28 Factor expressions in x and y. 2 2 (a) 4x − 9y ▯ Qishen Huang 5 CHAPTER 2. ALGEBRAIC EXPRESSIONS: BASIC 2 2 (b) x − 25 − 2xy + y (c) xy + 4xy + 4x (d) 4x 2− y + 2x + y 2 2 2 2 2 (e) (x + y ) − 4x y 3 2 (f) x (x − y) + x (y − x) 2 2 2.29 Multiply (x − y)(x + xy + y ). 2.30 Expand (a + b − c) 2. 2.31 Multiply (2a + b)(2a − b). 2.32 State the number of terms in expanded (a + b)(b + c). 2.33 Find Greatest Common Factors (GCFs) and Least Common Multipliers (LCMs). (a) x − 1 and x + 1 2 (b) x − 1 and x + 1 2 3 (c) x − 1 and x + 1 2.34 Find quotients and remainders. 3 2 (A) (x + x ) ÷ (x + 1) (B) (x 6+ x + x + x + x ) ÷ (x + 1)2 (C) (ax + 1) ÷ (x + 1) (D) (ax + bx + cx + d) ÷ (x − 1) 2.35 Complete squares in x in the expressions. (a) x − 2x − 3 2 (b) −2x − 8x − 9 2.36 Complete squares in x and y in expression x − 2x − 3y + 4y − 5. 2.37 Solve equation |x + 1| = 0. 2.38 Given equation |x − y| + |y − z| = 0, identity true statements about the variables. (a) All variables are zero. (b) All variables are equal. (c) Exactly two variables are equal. (d) At least two variables are equal. ▯ Qishen Huang 6 Chapter 3 Algebraic Expressions: Intermediate 3.1 Equation (−3a b 2n−1 )(3a1+nb ) = −9a b 4 is true for all a and b and some unknown constants m and n. Find the values of m and n. 2 3.2 Find the minimum value of 1 + 3(3 − x) . 3.3 Find the expression that is always greater: a + 2a + 4 a4+ a + 1 and . 3 4 3.4 Suppose a < −2. Simplify expression |2 − |1 − a||. 3.5 Find the minimum values of the two expressions. (a) |x − 3| (b) |x − 3| + |5 − x| 3.6 Compute 99999 + 199999 without a calculator . 2 3.7 Compute 2008 − 64 without a calculator. 3 3.8 Compute 777 − 776 × 777 × 778 without a calculator. 3.9 Evaluate the value of x + 2x + 1 at x = 9999 without a calculator . 3.10 If 3a + 5b = 9, compute the value of 1.5a + 2.5b + 0.5. 2 2 3.11 Solve equation (x − 78) = (x − 98) . 3.12 Compute the sum of the roots of equation (x − 1)(x + 9)(x − 5) = 0. ▯ Qishen Huang 7 CHAPTER 3. ALGEBRAIC EXPRESSIONS: INTERMEDIATE 2 4 b 3.13 Values a and b satisfy (a + 1) + (b − 23) = 0. Compute the value of a . 3.14 Given a − b = 1, compute the value of a − 3ab − b . 3 2 4 3 2 3.15 Given x +x−3 = 0, compute the value of x +2x +x without solving the equation. 3.16 Suppose x + y = 6 and xy = 4. Find the value of x y + xy without solving the equations. 2 2 3 3 3.17 Suppose a + b = 1 and a + b = 2. Find the value of a + b without solving the equations. 3.18 Define binary operation ▯ as follows: ( 2 a b, a ≥ b; a ▯ b = 2 ab , a < b. Solve equation 3 ▯ x = 48. 3.19 Suppose (x − a)(x + 2) = (x + 6)(x − b) is true for all x ▯ R. Find the values of a and b. ▯ Qishen Huang 8 Chapter 4 Rational Expressions 4.1 Solve for x in equation 1 2 3 x = 3+ 2. 4.2 Simplify expression (x + Δ) − x 2 Δ . 4.3 Simplify expression 1 1 x + Δ − x . Δ 4.4 Find the values of A and B that satisfy 1 A B x 2− 6x + 5 = x − 5+ x − 1. (The right hand side is called partial fractions of the left hand side.) a 4.5 List all possible values of, where a▯=0. |a| 4.6 Suppose ab < 0. Compute all possible values ofa + b . |a| |b| a b c 4.7 Find all possible values |a|+ |b|+ |c| where abc▯=0. a b a 4.8 Suppose b ▯= 0 and = . Compute the value of fraction. 3 5 b ▯ Qishen Huang 9 CHAPTER 4. RATIONAL EXPRESSIONS a b c a + b − 2c 4.9 Suppose 3 = 5 = 7 , where b▯=0. Compute the value of b . (2a − 2b)6 4.10 Simplify expression , where a ▯= b. (b − a) 2x − 9999 4.11 As x goes to infinity, what value does approach? 2x y 5x 4.12 Simplify expression + + 1. 5x − y y − 5x x + x2+ x + x 4 4.13 Simplify expression x−1 + x−2 + x−3 + x−4 . 2 √ 4.14 Evaluate expression x − 3 ÷ x − 2x − 3 at x = 5 + 1. x2 − 1 x2 + 2x + 1 x 2 √ 4.15 Evaluate expression x + at x = 3 + 1. x − 1 4.16 Suppose xyz ▯= 0 and 1 x x 1 = z . y Express z in terms of x and y. 4.17 Identify true identities. (a) x2 ≥ x. 1 (b) x ≥ , where x ▯ 0. x (c) x2 ≥ 2x − 1. 1 1 . xy 4.18 Simplify expression + 2 2. x − y x + y x − y x 4.19 If we increase x and y by 10%, by what percent does change? x + y (−3a 3)2 4.20 Simplify expression 2 . a 3 4 4.21 Simplify expression (−2x) + 6x − 12x . 3x 2 (x − y)2 − (x + y)(x − y) 4.22 Simplify expression . 2y ▯ Qishen Huang 10 CHAPTER 4. RATIONAL EXPRESSIONS 2 1 − x x + 1 4.23 Simplify expression x − 3x + 2x + x2 + x. 4.24 Observe 1 22 1 32 1 + = and 2 + = . 3 3 4 4 Find the general pattern. 4.25 If y ▯= 0 and 2x = 7y, compute the ratio of x : y. x y xy 2+ yx 2 4.26 If = , compute the value of 3 3 . 2 3 x + y x 3 x 4.27 Given y(x + y) =▯ 0 and x + y = 5 , compute the value of y. 4.28 Given a : b = 4 : 7, identify true statements. (A) (a + 1) : (1 + b) = 5 : 8 (B) (b − a) : b = 3 : 7 2 2 x − y √ √ 4.29 Evaluate the value of x y + xy2 at x = 5 + 1 and y = 5 − 1. 4.30 Polynomials f(x) and g(x) each have at least two unlike terms. Can f(x)g(x) and f(x) be a monomial? g(x) 4.31 Compute without a calculator: 123 ∙ 370 −123 . 122 ∙ 369 + 123 4.32 Given 1 1 1 1 = = = , a − 100 b + 101 c − 102 d + 103 sort a, b, c, and d in descending order. 4.33 Set {a, b, c} = {1234, 4567, 7890}. Choose the values of a, b, and c to minimize the value of 1 a + 1 1 b + c without computing any of its possible values. ▯ Qishen Huang 11 CHAPTER 4. RATIONAL EXPRESSIONS 4.34 Find the number of integer values of x that satisfy inequality 1 4 1 > > . 3 x 5 n n + 1 4.35 Prove inequality < , where n ≥ 1. n + 1 n + 2 2 2 1 1 4.36 If x + y = 2 and x2 + y2 = 1, find the value of |xy| without solving the equations. 1 1 4.37 If a + = 5, compute the value of a+ . a a2 ▯ Qishen Huang 12 Chapter 5 Linear Relations: Basic 5.1 Express the set of points in the fourth quadrant. 5.2 Point (a, b) is in the 2nd quadrant. Find the quadrants the following points are in. (A) (−a, b) (B) (a − 1, b) (C) (a, b + 1) (D) (1 − a, −b − 2) 5.3 Given set A on the graph, identify the set S defined by S = {(x,−y)|(x, y + 1) ▯ A}. y A D x B C 5.4 Line l is perpendicular to the y axis. Points P(1, 2) and Q are on line l. Find the y coordinate of Q. 5.5 Find where the lines x = 1 and y = −1 intersect, if at all. 5.6 Find the slope of a line connecting two given points in each case. ▯ Qishen Huang 13 CHAPTER 5. LINEAR RELATIONS: BASIC (a) (0, 0) and (0, 1) (b) (0, 0) and (1, 0) (c) (0, 0) and (1, 1) (d) (−1, −1) and (1, 1) 5.7 Determine whether the three points are coline (on a single line): A(1, 3), B(−2, 0), and C(2, 4). 5.8 Find the slopes of the following lines. (A) A horizontal line. (B) A vertical line. (C) y = x + 1. (D) y = −x + 1. (E) 2x + 3y + c = 0, where c is some constant. 5.9 In each case, determine whether line AB is parallel or perpendicular to CD. (a) A(0, 0), B(0, 1), C(0, 0), D(1, 0). (b) A(0, 1), B(2, 3), C(4, 5), D(6, 7). 5.10 Given two points P(5, 1) and Q(8, 9), perform the following tasks. (a) Write the equation of the line passing the two points. (b) Express the set of all points on the line segment between P and Q. (c) Express the set of all points on the line passing P and Q. (d) Express the set of points on the line passing P and Q and in the first quadrant. 5.11 Find the coordinates of the midpoint M between two points P(0, 1) and Q(2, 3). 5.12 Find the two points that trisect the line segment between two points (1, 6) and (10, 21). 5.13 Given point P(1, 2), find point Q that is symmetric to P with respect to each of the following axes and points. (a) The x axis. (b) The y axis. (c) The origin. (d) Point W(5, 6). ▯ Qishen Huang 14 CHAPTER 5. LINEAR RELATIONS: BASIC 5.14 Given points P(1, 6) and Q(8, 9), express the line segment symmetric to segment PQ with respect to each of the following axes and points. (a) The x axis. (b) The y axis. (c) The origin. (d) Point R(5, 6). 5.15 Determine the sign of the slope of the line that isn’t any axis in each case. (a) It is not present in the 3rd and 4th quadrants. (b) It is not present in the 2nd and 4th quadrants. (c) It is not present in the 2nd and 3rd quadrants. 5.16 Check whether the following points are on line y = 8x + 9. (a) (0, 1) (b) (0, 9) 5.17 How many points are needed to determine a line? 5.18 Are the two equations equivalent? (a) x + 2y = 9 (b) 2x + 4y − 18 = 0 5.19 Line l intercepts the x axis at 4 and the y axis at 5. Write its equation. x y 5.20 Write the two following equations in intercept form + = 1. a b (A) x + 2y = 3 (B) 4x − 5y = −6 5.21 Write an equation in slope-intercept form for the line with slope of 3 and y intercept of 2. 5.22 Write the following equations in slope-intercept form y = kx + b. (A) x + 2y = 3 (B) 4x − 5y = −6 x y 5.23 Find the slope of linear equation + = 1, where a and b are some non-zero constants. a b ▯ Qishen Huang 15 CHAPTER 5. LINEAR RELATIONS: BASIC 5.24 Find the intercepts of linear equation y = kx + b, where k ▯= 0. 5.25 Write an equation for the line l that passes point (1, 2) and is parallel to each line below. (a) y = 2x + 1 (b) y = 6 (c) x = 8 5.26 Write an equation for the line l that passes point (1, 2) and is perpendicular to each line below. (a) y = 2x + 1 (b) y = 6 (c) x = 8 5.27 Write equations of the lines passing the origin. 5.28 Compute the distance between the two points in each pair. (a) (0, 1) and (2, 3) (b) (1, 0) and (2, 0) (c) (0, 3) and (0, 4) (d) (−1, −1) and (1, 1) 5.29 Find the distance from point P(2, −3) to each axis. (a) The x axis. (b) The y axis. 5.30 Compute the area of the triangle formed by the x axis, the y axis, and line y = −x+2. 5.31 Find the distance between two parallel lines: x = 6 and x = −8. 5.32 Find the area of a triangle with vertices of A(0, 0), B(3, 4), and C(4, 3). 5.33 Find the value of k such that line y = kx + 3 passes point (1, 2). 5.34 John drives at constant speed of 40 miles per hour. Express the distance he travels as a function of time. 5.35 Solve equation |2x + 7| = 1. 5.36 Solve two equations in two variables. ▯ Qishen Huang 16 CHAPTER 5. LINEAR RELATIONS: BASIC (a) |x − 1| + |x + y| = 0 (b) (x − 1) + y = 0 5.37 Positive integers a and b have no common factor except 1. In addition, b = 5(a − b). Find the values of a and b. ▯ Qishen Huang 17 Chapter 6 Linear Relations: Intermediate 6.1 Find line l that is half way between lines m: y = kx + a and n: y = kx + b, where k ▯= 0 and a ▯= b. 6.2 Find the distance between two parallel lines: y = 2x + 5 and y = 2x + 10. 6.3 Find the distance from point P(2, 3) to line l: x + 2y = 3. 6.4 Find the point Q on line l: x − 2y = 1 that is closest to point P(1, 1). 6.5 Write an equation for the line l that is symmetric to line m: y = 2x + 1 with respect to each axis. (a) The x axis. (b) The y axis. 6.6 Show that line ax+by = c is symmetric to bx+ay = c with respect to line x−y = 0. 6.7 Solve equation |2x + 8| = |4x − 1|. 6.8 If 71 b 5 × 17 ,bfind the value of base b. 6.9 Function f(x) = 6x + 8. Perform the following tasks. (a) Find the inverse function of f. (b) Determine whether the inverse function f−1 is linear. −1 (c) Compute the product of the slopes of functions f and f . −1 (d) Show that f is symmetric to f with respect to line y = x. 6.10 Functions f(x) = Ax + B and g(x) = Cx + D, where A, B, C, and D are constants and AC ▯= 0. Identify linear functions on the following list. (a) 2f(x) + 6g(x) ▯ Qishen Huang 18 CHAPTER 6. LINEAR RELATIONS: INTERMEDIATE (b) f(x)g(x) (c) f(x) + 8 (d) |f(x)| (e) f(g(x)) (f) 2f−1 (x) + 6g(x) −1 6.11 A linear function f has slope of 1. In addition, f (2) = 3. Find the value of f(4). 6.12 Solve equation 5x − 6y = 0, where x and y are positive integers less than 10. 6.13 Define function f(x) as follows: ( x + 1, x ≤ 1; f(x) = 3 − x, x > 1. Evaluate f(f(2)). 6.14 Given linear function f(x) = ax + b, define g(x) as follows: g(x) = f(x + 1) − f(x). Write g in algebraic form. 6.15 Point (x ,0y 0 is not on line l: f(x, y) = 0. Define line m as follows: f(x, y) − f(x , y ) = 0. 0 0 Answer two questions. (a) Is point (x , y ) on line m? 0 0 (b) Are the two lines parallel or perpendicular? 6.16 Write an equation of a plane with x intercept of 1, y intercept of 2, and z intercept of 3. 6.17 Define set S as S = {(x, y)|2x + y ≤ 4, x − y ≤ 1, x ≥ 0, y ≥ 0}. Mark the set on the graph. ▯ Qishen Huang 19 CHAPTER 6. LINEAR RELATIONS: INTERMEDIATE y 2x + y = 4 x − y = 1 x ▯ Qishen Huang 20 Chapter 7 Linear Relations: Advanced 7.1 Function f(x) = 3x + 4. Perform the following tasks. (a) Give the domain and range of the function. (b) Determine its monotonicity (increasing or decreasing). (c) Prove that for any 1 and x2in the domain, f(ax1+ (1 − a)x 2 = af(x 1 + (1 − a)f(x2), for any a ▯ R. 7.2 Function f(x) = ax+b. Point C(x ,C0) is the midpoint between points A(x A 0) and B(x B 0). Prove that (xC, f(x C) is the midpoint between points (xA, f(xA)) and (xB, f(xB)). 7.3 Points P(x 1, y1) and Q(x2, y2) are above line l: y = ax + b. Prove all points in the set S = {(kx1+ (1 − k)x 2 ky1+ (1 − k)y2), k ▯ (0, 1)} are above the line. 7.4 Show that the graph of equation xy + y = 0 consists of two lines. 7.5 Graph the solution set of inequality 2 2 x y + xy + xy ≥ 0. 7.6 Given points P(0, 0) and Q(2, 2), write an equation for all points A(x, y) such that |PA| = |QA|. 7.7 Solve equation |x − |x − 3|| = 5. 7.8 Solve equation |x − |x − |x − 3||| = 5. ▯ Qishen Huang 21 CHAPTER 7. LINEAR RELATIONS: ADVANCED 7.9 Suppose x − y + 0.5z = 10 and x + 3y + 1.5z = 90. Find the value of x + y + z. 7.10 Given an arbitrary equation f(x, y) = 0, find another equation that is symmetric to the equation with respect to line x + y − 2 = 0. 7.11 Constants k and k satisfy k k = 1. Find the axes of symmetry about which the 1 2 1 2 line y = k1x is symmetric to y = k 2. 7.12 Use techniques in linear equations to express 0.5168 as a fraction number. 7.13 Regardless of the value of parameter m, line (m + 3)x + (2m − 1)y + 7 = 0 passes a fixed point. Find the point. ▯ Qishen Huang 22 Chapter 8 Word Problems: Basic 8.1 An 8 inch pizza is cut into 4 equal slices. A 10 inch pizza is cut into 6 equal slices. Which slices are larger? 8.2 A shirt is on sale at a 20% discount. The sales tax is 5%. A new sales clerk simply takes 15% off the original price. Is this correct? Why? 8.3 Twenty seven white unit cubes are stacked into one large cube. The surface of the large cube is then painted red. After the painting, how many small cubes are still completely white? 8.4 If the general inflation rate was 3% last year and John’s salary grew 4%, how much did the purchasing power of his salary change? 8.5 Find the speed in degrees per minute of the minute hand of an accurate a clock. 8.6 At an office supplies store, pencils sell individually for $0.10 each and in packs of 12 for $0.80 per pack. Kay buys 2 packs instead of 24 individual pencils. How much money does she save? 8.7 The price of regular rice is $1 per pound; that of premium rice is $2. Mix 300 pounds of regular rice with 400 pounds of premium rice. What is the price of the mixture? 8.8 A car runs 30 miles per gallon of gasoline. How long can it run with $40 worth of gasoline purchased at $4 per gallon? 8.9 Mary and Jane shop at an office supplies store. Mary spends $20 on 10 pens and 10 notebooks. Jane spends $15 on 5 pens and 8 notebooks. Find the prices of the stationery. 8.10 A group of friends share cost of their party. If each pays $42, there is surplus of $19. If each pays $40, the total is short of $8. How much does the party cost? ▯ Qishen Huang 23 CHAPTER 8. WORD PROBLEMS: BASIC 8.11 Dad is 42 years old and his son is 10. In how many years will Dad be 3 times as old as his son? 8.12 A sign on a freeway says 1 mile to the 15th Street exit and 4 miles to the Downtown exit. At a different location on the same road in the opposite direction, another sign says 2 miles to Downtown exit. How far is the sign from the 15th Street exit? 8.13 John is twice as old as her sister Jane. Four years ago, John was three times as old. What is John’s current age? 8.14 The price of a stock is down 5% today. What percent increase tomorrow would return the stock to its original price? 8.15 A product is currently priced at $10. If it is discounted by 10%, the profit margin will be 10%. What is the cost of the product? 8.16 A pair of shoes costs $32 after a 20% discount. What is the original price? 8.17 A closing factory sold two used machines for $1200 each. One machine brings 20% profit; the other 20% loss. How much is the total profit or loss? 8.18 Two toy stores, A and B, sell Game Boys. The regular price at store A is 80% of the manufacturer’s suggested retail price. Store B normally does not discount at all. During Christmas season, store A discounts the product by 20% off its regular price. Store B discounts it by 40% off its regular price. Which store offers a better deal? 8.19 A store bought 10 vases for $15 each from supplier A. It bought 40 vases for $12.5 each from supplier B. Its profit margin is supposed to be 10%. How much should the selling price for each vase be? 8.20 Johnny has $100 now and saves $10 every month. His brother Jimmy has no money currently and is going to save $15 per month. In how many months will Jimmy have as much money as Johnny? 8.21 A football team has won 10 games and lost 5. If the team wins the remaining games, they will have won 75% of all the games. How many more games will they play? 8.22 A baseball team won 20% of its games in the first half of the season. To achieve 50% winning average or better in the season, at least what percent of its remaining games must it win? 8.23 A pizza restaurant wants to make a pizza that is 10% bigger than the current 8 inch pizza. What is the diameter of the new pizza? 8.24 Four pizzas are ordered for the children at a party. Each pizza is cut into 8 equal slices. Each child eats 2 slices. There are 2 slices left. How many children are at the party? ▯ Qishen Huang 24 Chapter 9 Word Problems: Intermediate 9.1 A mother horse and a baby horse walk 10 miles from point A to point B. The mother’s speed is 10 miles per hour and the baby’s 5 miles per hour. The mother rests for 2 minutes after every x minutes of walking. The baby keeps walking. They arrive together at the destination. Find the range of x. 9.2 A worker in a sunglass factory can make 50 frames or 100 lenses per day. There are 90 workers. How many workers should make lenses? 9.3 A store bought a number of tractors for $800 each. Its regular selling price is $1000. The store decides to discount the price to stimulate sales and to earn a minimum profit of 5%. What is the maximum discount rate? 9.4 Kate took a math test that she missed due to sickness. Her perfect score of 100 points raises class average from 80 points to 81. How many students including her in the class took the test? 9.5 John drives from point A to point B at a constant speed. On his way back, he drives 20% faster and spends 12 minutes less. How much time does he spend on the round trip? 9.6 An accurate clock shows exactly 3 pm. In how many minutes will the minute hand catch up with the hour hand? 9.7 A nut mix contains 10% of peanuts and 90% of cashews. It costs 5% more than pure peanuts. Are cashews more expensive than peanuts? By how much? 9.8 A mobile phone company offers two monthly calling plans, A and B. Under A, every minute costs $0.10. Under B, the first 100 minutes costs $12 and every extra minute costs $0.08. What monthly talking time would make B the better choice? 9.9 It takes an escalator 20 seconds to move a person from the first floor to the second. If not operating, Jose takes 30 seconds to walk up on the still escalator. If Jose walks while the escalator is moving, how long does it take for him to reach the second floor? ▯ Qishen Huang 25 CHAPTER 9. WORD PROBLEMS: INTERMEDIATE 9.10 A project must be finished in 80 days. Contractor A costs $150 a day; Contractor B costs $100 per day. The project manager has three possible choices: (a) Hire contractor A alone. The project would be finished just in time. (b) Hire contractor B alone. But it’d take 100 days. (c) Mix use of contractor A and contractor B. Find the solution to minimize total cost and to meet deadline. 9.11 A team of 4 people has just finished half of a project in 10 days. If the rest of the project needs to be done in 5 more days, how many more people are needed? 9.12 It takes Mary 6 days to complete a task. It takes Mike 8 days to do the same thing. If both work together, how long would it take? 9.13 Six people to can perform a task in 8 days. If we add 2 equally able people, how long would it take? 9.14 It takes 9 people 9 days to finish a project if they work 9 hours a day. If only 8 people work only 8 hours a day, how long would it take? 9.15 A 20 lb bag of nut mix has 40% of peanuts and 60% of cashews. To get 20% peanuts, how much of the mix should be replaced with pure cashews? 9.16 Bob walks from point A to point B at 6 miles per hour and back at 8 miles per hour. What is his average speed (total distance divided by total time)? 9.17 Tom drives 500 miles on the first day. On the second day, he drives twice as long and his average speed is 4/5 of that on the first day. How long does he drive on the second day? 9.18 On the eve of the Democratic Illinois primary, a poll finds 55% of the voters support candidate A, and 24% support B. Exactly 20% are undecided now and will support someone. Estimate the percentage of voters who will eventually support A. 9.19 Mary and Jane work part time for a library. Mary works every third day; Jane works every fifth day. Today, they are both working together. In how many days will they be working together again? ▯ Qishen Huang 26 Chapter 10 Word Problems: Advanced 10.1 Kay spends $134 on 11 tickets of two different classes. Class A tickets are $3 more expensive than class B. How many class A tickets and at what price does she buy? 10.2 Two classes took a test. Class A averaged 80 and class B averaged 90. The two classes together averaged about 83. Which class has more students? 10.3 In a family of five, members speak English, Spanish, or both. Two people speak Spanish, and four English. How many people speak both languages? 10.4 Students of a high school take a state assessment test. A class passes the test if half or more of the students pass. Which is greater: the percentage of passing classes or the percentage of passing students? 10.5 A Math quiz has 25 questions. A correct answer earns 8 points, and a wrong one costs 3 points. Jess scores 110 points. How many questions does Jess answer? 10.6 A man wants to see his girlfriend who lives 6 miles away. He leaves his house for her house at 6 am and walks at 4 miles an hour. When he leaves, he also sends a pigeon to her house. The bird flies at 30 miles an hour. When the bird reaches her house, she walks toward him at 2 miles an hour. When will they meet on the way? 10.7 An electric power company charges its consumers $0.40 per kilowatt for the first 100 kilowatts in a month, $0.50 per kilowatt for the second 100 kilowatts, and $0.60 for each additional kilowatt beyond. Mary pays $120 for electricity used in last month. How much electricity did her family consume in the month? 10.8 It takes Jose 12 days to complete a task. The same task would take Hans 24 days. After the two men work on the task for 4 days, Jose leaves for vacation and Hans continues. In how many more days will Hans finish the work? ▯ Qishen Huang 27 CHAPTER 10. WORD PROBLEMS: ADVANCED 10.9 Sergei makes a round trip from home to the library. The distance from home d (in meters) is a function of time t (in minutes): 50t, 0 ≤ t ≤ 20; d(t) = 1000, 20 < t < 30; 2500 − 50t, 30 ≤ t ≤ 50. How far is the library from his home? How long does he stay in the library? How fast does he walk home? 10.10 Jim and John drive from point A to point B in separate cars. Jim leaves at 6 am and arrives at 4 pm. John leaves at 10 am and arrives at 3 pm. Assume both men drive at constant speeds. Find when John catches up with Jim. 10.11 John and Bill walk towards to each other’s house. If both leave at 10 am, they meet 10 minutes later. If Bill leave 3 minutes later then John, they’ll meet 9 minutes after Bill leaves. How long does it take for Bill to walk to John’s house? 10.12 Wang starts a project, which he can finish in 20 days. Five days later after he starts, his company sends Lee to help. Together, they finishes the project in 5 more days. If Wang is paid $100 a day, what is a fair daily salary for Lee? 10.13 Todd agreed to work for his Dad for 20 days to get $500 and an iPod. He worked for only 10 days and had to quit for some reason. He got $200 and the iPod based on the amount of work done. How much was iPod valued? 10.14 A clock is broken. But its hour hand, minute hand, and second hand still move at constant speeds in the normal direction. The second hand passes the hour hand every 10 seconds (measured by an accurate clock) and passes the minute hand every 20 seconds. How often does the minute hand pass the hour hand? 10.15 In a group of tennis players, a person plays either single or mixed double, but not both. 1/2 of the men and 1/3 of the women play mixed double. What fraction of the group play mixed double? ▯ Qishen Huang 28 Chapter 11 Geometry: Basic 11.1 Answer the questions regarding points and lines. (a) Point P is not on line l. How do you find a point on the line so that the two points have the shortest distance? (b) Point P is not on line l. How many lines passing P are parallel to l? (c) Point P may and may not be on line l. How many lines passing P are perpen- dicular to l? 11.2 Identify true statements. (a) A point in geometry has no size. (b) A line in geometry has no length, direction, or thickness. (c) Two congruent triangles are similar. (d) Corresponding altitudes of two congruent triangles are equal. 11.3 Identify true statements. (a) The ratio of a pair of corresponding altitudes of two similar triangles is equal to the ratio of any pair of corresponding sides. (b) The ratio of the areas of two similar triangles is equal to the ratio of a pair of corresponding sides. (c) All isosceles right triangles are similar. (d) All equilateral triangles are similar. 11.4 Identify true statements about ▯ABC. (1) If ▯A > B, then a > b. (2) If a > b, then the altitude on side a is shorter than that on b. ▯ Qishen Huang 29 CHAPTER 11. GEOMETRY: BASIC 11.5 Two quadrilaterals ABCD and A B C D sati1fy 1 1 1 ▯ A = A , 1 ▯ B = B , 1 ▯ C = C , a1d ▯ D = D . 1 Are the quadrilaterals similar? 11.6 Two quadrilaterals ABCD and A 1 B 1 D1sa1isfy AB = A 1B 1 BC = B C , 1D 1 C D , and D1 =1D A . 1 1 Are the quadrilaterals congruent? 11.7 Answer the questions regarding points, tangent lines, and circles. (a) Point P is outside circle C. How many tangent lines of the circle pass the point? (b) Point P is outside circle C. How would you find the point on the circle that is closest to P? (c) Two circles of different radii do not intersect. How would you find one point on each circle such that the two points have the longest distance among similar points? (d) A line and a circle do not intersect. How would you find one point on the line and another on circle such that the two points have the shortest distance among similar points? 11.8 Find the measure of the angle that the second hand of an accurate clock rotates in 20 seconds. 11.9 State the sum of all angles of a triangle and that of a quadrilateral. 11.10 Find the polygons below that are symmetric with respect to some point. (a) Parallelogram (b) Equilateral triangle (c) Rhombus (d) Trapezoid 11.11 Find the number of axes of symmetry of the following shapes. (a) Equilateral triangle (b) Square (c) Rhombus but not a square (d) Regular hexagon (e) Circle ▯ Qishen Huang 30 CHAPTER 11. GEOMETRY: BASIC 11.12 Given ▯ABC with each condition below, determine whether it is a right triangle in each case. (1) The ratio of three angles is 1 : 2 : 3. (2) The ratio of three sides is 3 : 4 : 5. 2 (3) a = (c + b)(c − b). 11.13 A square and a circle have equal areas. Which shape has shorter perimeter? 11.14 Compute the area of a triangle with three sides of 6, 8, and 10. ◦ 11.15 A right triangle has an acute angle of 30 and a hypotenuse of 1 unit long. Find the lengths of the two other sides. 11.16 Two sides of a right triangle are 1 unit long. Find the length of the third side. 11.17 The three sides of an equilateral triangle are 1 unit long. Find the altitudes of the triangle. 11.18 What is circumcenter of a triangle? Is it inside the triangle? In ▯ABC, the per- pendicular bisectors of sides AB and BC intersect at point D. Connect D with the midpoint E of side AC. Find the measure of ▯AED. 11.19 What is centroid of a triangle? In ▯ABC, lines AD and BE each divide the triangle into two equal areas. The two lines intersect at point F. Connect points C with F and extend the line segment to divide the triangle into two areas. Find the ratio of the two areas. 11.20 What is the orthocenter of a triangle? Is it inside the triangle? ▯ABC has area of 10. Lines AD and BE are altitudes on sides BC and AC respectively. AD and BE intersect at point F. Connect points C with F and extend the line segment to intersect side AB at point G. Compute product CG × AB. 11.21 What is the incenter of a triangle? Is the center equally distant from the three sides or three vertices of the triangle? 11.22 Find how many diagonals can be drawn from a vertex of convex n-sided polygon (n > 3). 11.23 State the number of diagonals in a quadrilateral and that in a hexagon. 11.24 Find the sum of all angles in a n-sided polygon (n ≥ 3). ◦ 11.25 Every interior angle of a huge regular polygon is 172 . How many sides does the polygon have? 11.26 If an interior angle of a polygon is 45 , what is the corresponding exterior angle? ▯ Qishen Huang 31 CHAPTER 11. GEOMETRY: BASIC 11.27 Compute the sum of all exterior angles of a polygon. 11.28 At least how many acute angles does a triangle have? At most how many acute angles does a quadrilateral have? 11.29 If all interior angles of a polygon are equal, are all the sides equal? 11.30 In ▯ABC and ▯A B C , 1 1 1 AB = 2, BC = 3, AC = 4, A 1B 1 4, B C 1 1, and A C = 81 1 The altitude on side AB is CD, and that on side A B is1C 1 . Fi1d 1he ratio of CD : C D1. 1 11.31 If the base of a triangle increases by 5% and its corresponding altitude decreases by 5%, by what percent does the area of the triangle change? 11.32 Lengthening the three sides of a triangle by 100%, we get a new larger triangle. By what percents do its three angles, perimeter, and area change? 11.33 The shadow of a 1-meter tall tree is 0.4 meter long. If the shadow of another tree is 2 meters, how tall is the tree? 11.34 On the graph, each grid is 1 × 1. Compute the area of ▯ABC. C A B 11.35 The central angle of a sector of a unit circle is 50 . Find the area of the sector. 11.36 A sector of a unit circle has perimeter of 2.5. Find the area of the sector. 11.37 Two circles do not intersect and are outside each other. How many tangent lines can they have in common? 11.38 Two circles of radii 1 and 2 intersect at only one point. How many tangent lines can they have in common? 11.39 A unit circle is inscribed in ▯ABC. In the triangle, ▯A = 60 . Connect the circle center O with vertex A. Find the length of segment AO. ▯ Qishen Huang 32 CHAPTER 11. GEOMETRY: BASIC 11.40 Point A is outside a circle. The longest distance between the point and the circle is 9 and the shortest 5. Find the radius of the circle. 11.41 The two diagonals of a parallelogram intersect at point A. Find the minimum angle the shape rotates around A to coincide with the original shape. 11.42 Prove that the area of a rhombus is half of the product of its two diagonals. 11.43 Draw two diagonals of a parallelogram (not a rhombus or rectangle). Find the number of pairs of congruent triangles in the parallelogram. 11.44 An angle of an isosceles triangle is twice as big as another. Find the measure of the smallest angle of the triangle. 11.45 Three sides of a right triangle are 3, 4, and 5. Find the altitude on the hypotenuse. 11.46 Every side of a regular hexagon is 1. Find the distance from the center to any side. 11.47 The sides of ▯ABC are a, b, and c. Determine the sign of expression a − 2ab + b − c . 11.48 ▯A of ▯ABC is greater than the corresponding exterior angle. Is the triangle acute, right, or obtuse? 11.49 In right ▯ABC, BD is the altitude on side AC. Find the number of pairs of similar triangles in ▯ABC. C D A B 11.50 If the two diagonals of a quadrilateral are perpendicular to each other, is the polygon a rhombus? 11.51 In ▯ABC, points D and E are the midpoints of sides AB and AC respectively. In addition, BC = 10. Find the length of segment DE. 11.52 The three midsegments of a triangle form another smaller triangle. The perimeter of the smaller triangle is 10. Find the perimeter of the original triangle. ▯ Qishen Huang 33 Chapter 12 Geometry: Intermediate 12.1 Three distinct points are on a plane. Draw a line through every pair of two points. How many distinct lines are obtained? 12.2 One hundred lines, l 1 l2, 3 , ...,100 satisfy l1▯ l2, l2▯ l3, ∙∙∙ , 99▯ l100. Is line1lparallel or perpendicular to 100 12.3 A couple of unit circles overlap partially. Find the perimeter of the non-overlapping region. 12.4 Two chords AB and CD of a circle intersect at point P inside the circle. Prove that AP ∙ PB = CP ∙ PD. ▯ Qishen Huang 34 CHAPTER 12. GEOMETRY: INTERMEDIATE B C P D A 12.5 Three angles of an isosceles triangle are 70nd 40 . The base of the triangle is a diameter of a circle. Find the measure of the minor arc of the circle that the two equal sides intercept. 12.6 The area of ▯ABC is 25. Point D is on side BC such that BD = 4 and DC = 6. Find the area of ▯ABD. A B D C 12.7 Connecting clockwise the midpoints of the four sides of a quadrilateral, we get a smaller quadrilateral inside. Prove the new quadrilateral is a parallelogram. ▯ Qishen Huang 35 CHAPTER 12. GEOMETRY: INTERMEDIATE 12.8 In parallelogram ABCD, point E on side AB and point F on side CD satisfy AE = CF. Connect point D with E, and B with F. Prove that quadrilateral BFDE is a parallelogram. D F C A E B 12.9 Each side of a rhombus is 5. One of its diagonals is 8. Find the radius of the inscribed circle. 12.10 An equilateral triangle has an inscribed circle and a circumscribed circle. Find the ratio of their radii. 12.11 Three unit circles are mutually tangent. Find the perimeter of the gap area that the circles enclose. ▯ Qishen Huang 36 CHAPTER 12. GEOMETRY: INTERMEDIATE 12.12 The radii of two concentric circles are 5 and 2. A chord of the bigger circle is tangent to the small circle. Find the length of the chord. 12.13 A quadrilateral has an inscribed circle. In addition, the sum of the top side and the bottom is 8. Find the perimeter of the quadrilateral. 12.14 In ▯ABC, ▯B = 80 . The bisectors of the two exterior angles at A and C intersect ▯ at point D. Find the measure ofADC. E D C F A B 12.15 The perimeter of a triangle is an odd integer. In addition, its two sides are 2 and 9. Find all possible values of the third side. ▯ Qishen Huang 37 CHAPTER 12. GEOMETRY: INTERMEDIATE 12.16 Two diagonals of a rhombus are equal. Is it a square? 12.17 The two diagonals of a rectangle ABCD intersect at point E. IAEB =ition, 120 . Find the measureADE. D C E A B 12.18 The sides of ▯ABC are a, b, and c. Show that |a − b| < c. 12.19 The diagonals of a quadrilateral are two diameters of its circumcircle. What is the shape of the quadrilateral? 12.20 Two equilateral triangles have sides of 0.5 and 3 units long. How many of the smaller triangles are needed to fill the bigger triangle without overlapping? 12.21 ▯ABC has area of 16. Three lines parallel to side BC divide side AC into four equal segments. As a result, three non-overlapping trapezoids are produced. Find the area of the middle trapezoid. A D F E G B C 12.22 Point E is a point inside unit square ABCD. Connecting E with the four vertices, we have four triangles. Find the sum of the areas of the top and the bottom triangle. D C E A B ▯ Qishen Huang 38 Chapter 13 Geometry: Advanced 13.1 In a convex polygon, at most how many interior angles are acute? 13.2 In quadrilateral ABCD,A = 60 and ▯D = 70 . Opposite sides AB and CD do not intersect. Find the measure of the angle formed by the their extensions. D C A B 13.3 In rectangle ADEF, AF = 4 and AD = 6. The bisectors of top two aE and ▯ F and the bottom side AD form a triangle. Find the area of the triangle. F E G A B C D 13.4 Each altitude of four congruent quadrilaterals is longer than 1. Try to cover a unit circle completely with the four such quadrilaterals without overlapping. ▯ Qishen Huang 39 CHAPTER 13. GEOMETRY: ADVANCED 13.5 Connecting clockwise the midpoints of adjacent sides of an isosceles trapezoid, we get a new smaller quadrilateral. What kind of quadrilateral is it? 13.6 Two intersecting circles have radii of 1 and 3. Find the range of possible distance between their centers. 13.7 A unit circle is inscribed in a trapezoid. Find the minimum of the perimeter of the trapezoid. 13.8 Two circles of radii 1 and 3 are outside each other and mutually tangent. Draw a common tangent line through the tangent point and another common tangent line. Find the measure of the angle included by the two lines. 13.9 A circle passes all four points that trisect the two diagonals of a unit square. Find the diameter of the circle. ▯ Qishen Huang 40 CHAPTER 13. GEOMETRY: ADVANCED 13.10 A star has five vertices. Find the sum of the angles at those vertices. 13.11 Given a rectangle and a point, find a line passing the point and dividing the rectangle into two regions of equal areas. 13.12 A square is just big enough to contains a unit circle. What is radius of the largest circle in one of the corners in the square but outside the unit circle? 13.13 ▯ABC has area of 1. Its side BC is 2. Line DE is parallel BC. If ▯ADE has area of 0.5, find the length of DE. A D E B C ▯ Qishen Huang 41 CHAPTER 13. GEOMETRY: ADVANCED 13.14 A circle is inscribed in a unit square, and a smaller square is inscribed in the circle. Find the area of the smaller square. 13.15 Two mutually perpendicular lines pass the center of a unit square and divide the square into four regions. Prove that the four regions have equal areas. 13.16 Find the altitude of the smallest cone in terms of volume that contains a unit sphere. 13.17 The center of a 5 × 5 square is at midpoint of one side of a 8 × 8 square. Find the size of the overlapping area. B A ◦ 13.18 We cut a circle sector of 120 from a unit circle. The remaining is used to form a cone. Find the radius of the base circle and the height of the cone. ◦ 13.19 Rotate a unit square clockwise around its center by 30 . Find the overlapping area of the new and the old square.

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