Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 7 Geometry
Problem
Solution
1) Triangle ABC has the following properties. When you place 5 congruent copies by rotating around the point B as shown on the left, angle ABP is a right angle. When you place 6 congruent copies by rotating around point A as shown on the right, a star is formed with no overlap at the center. What is the measure of angle C?
  1. The center angle, ABC is 1⁄5 of 90 degrees =
    18 degrees.
  2. Angle BAC is 1⁄6 of 360 degrees =
    60 degrees.
  3. C = 180 - 60 - 18 = 102 degrees
2) A can is designed to hold three balls with diameter 2 cm exactly as indicated in the diagram. The can is filled with water. Then the balls are pushed in one at a time keeping the can upright. How much water remains in the can when the balls are completely in the can? Give your answer to the nearest cubic centimeter or leave in your answer.
(Volume of a sphere: 4⁄3 r3)
    =3.14
  1. The height of the can is 3 widths of the balls = 6 cm.
  2. Volume of the can = 6 cu. cm.
  3. Volume of 1 ball = 4⁄3 cu. cm.
  4. Difference = 6 - 3 x 4⁄3 =
    2 cu. cm.

Problem
Solution
3) Rhonda wants to build a pendulum clock with the pendulum enclosed in a box with a glass door. Her pendulum will swing through 30 degrees and be 3 feet long. She's decided to use the length of the arc the pendulum swings through as the width for the box. What will the width be to the nearest tenth of a foot? (The arc is a section of the circumference determined by the center angle.)
Use = 3.14 or express your answer in terms of
  1. The circumference of the circle is
    6 ft.
  2. 30/360 of the circumference =
    (6)/ 12 = /2 = 1.6 ft.
4) The Fantastic Island Puzzle consists of colored pieces that are arrangements of 5 balls stuck together. One example is shown. One of the puzzles is to make a tetrahedron (triangular pyramid) whose base is an equilateral triangle that takes 4 balls along an edge as shown. How many pieces are needed to make the tetrahedron?
 Careful! They are asking for the number of 5-ball pieces NOT the number of balls!
  1. The base consists of 10 balls
  2. The next layers consists of 6, 3 and 1 balls for a total of 20 balls
  3. Each piece consists of 5 balls, so 20/5 =
    4 pieces are necessary.
5) A wooden triangular prism has two equilateral triangle faces with edge length 10 cm. The long edge of the prism has length 16 cm. Nigel is going to make a smaller prism by making a horizontal cut meeting the midpoints of the two legs of each triangle. What is the area of the new bottom rectangle? Figure not to scale. If the equilateral triangle is cut at it's midpoint, then the triangle remains equlateral with a side length of 5 cm.

The new bottom rectangle's dimensions are 5x16.
It's area = 80 sq. cm.