Mathworld.com Classification: Geometry > Plane Geometry > Miscellaneous Plane Geometry > Area Calculus and Analysis > Differential Geometry > Differential Geometry of Curves > Perimeter 18
Three circular arcs of radius 5 units bound the region shown. Arcs AB and AD are quarter-circles, and arc BCD is a semicircle. What is the area, in square units, of the region? A B C D (A) 25 (B) 10 + 5 π (C) 50 (D) 50 + 5 π (E) 25 π 2000 AMC 8, Problem #19— “Change the shape of the figure to a rectangle.” Solution Answer (C): Divide the semicircle in half and rotate each half down to fill the space below the quarter-circles. The figure formed is a rectangle with dimensions 5 and 10. The area is 50. I II I II OR Slide I into III and II into IV as indicated by the arrows to create the 5 × 10 rectangle. I II IV III Difficulty: Hard NCTM Standard: Geometry Standard for Grades 68: apply transformations and use symmetry to analyze mathematical situations. Mathworld.com Classification: Geometry > Plane Geometry > Miscellaneous Plane Geometry > Area 19
You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of 1.02, with at least one coin of each type. How many dimes must you have? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 2000 AMC 8, Problem #20— “Since the total value is 1.02, you must have either 2 or 7 pennies.” Solution Answer (A): Since the total value is 1.02, you must have either 2 or 7 pennies. It is impossible to have 7 pennies, since the two remaining coins cannot have a value of 95 cents. With 2 pennies the remaining 7 coins have a value of 1.00. Either 2 or 3 of these must be quarters. If you have 2 quarters, the other 5 coins would be dimes, and you would have no nickels. The only possible solution is 3 quarters, 1 dime, 3 nickels and 2 pennies. Difficulty: Medium-hard NCTM Standard: Algebra Standard for Grades 68: relate and compare different forms of representation for a relationship. Mathworld.com Classification: Number Theory > Arithmetic > Addition and Subtraction 20
Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is (A) 1 4 (B) 3 8 (C) 1 2 (D) 2 3 (E) 3 4 2000 AMC 8, Problem #21— “Make a complete list of equally likely outcomes.” Solution Answer (B): Make a complete list of equally likely outcomes: Same Number Keiko Ephraim of Heads? H HH No H HT Yes H TH Yes H TT No T HH No T HT No T TH No T TT Yes The probability that they have the same number of heads is 3 8 . Difficulty: Hard NCTM Standard: Data Analysis and Probability Standard for Grades 68: compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models. Mathworld.com Classification: Probability and Statistics > Probability 21
A cube has edge length 2. Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to: (A) 10 (B) 15 (C) 17 (D) 21 (E) 25 2000 AMC 8, Problem #22—
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