Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 7 Geometry
Problem
Hint
1) The net shown folds to a cube with a missing face. Mark with an x two edges where the final square can be placed so that the net will fold to a cube with no missing faces.
  1. Start from face #1 and start folding, keeping face#2 as the one facing you (be careful!)
  2. After 4 folds you should have your assembled 3-D cube.
  3. Mark at least 2 edges on the diagram to the left that border the missing face. Since this is a cube, there will be 4 faces that border the opening. You can choose 2 of them.
2) The point P (2, 3) is plotted for you. Reflect this point across y = x. Reflect P and its image across x = 0. Keep alternating reflecting the new image points across either y = x and across x = 0 until no new image points can be created. The points are the vertices of a polygon. How many lines of symmetry does the polygon have?
  1. Mark on this blank graph the initial point, P.
  2. Now draw the lines of the equations for the reflecting axes (y = x) and (x = 0)
  3. Perform the rotations, until no repetitions result from any further reflections and put those points on the graph.
  4. Identify the lines of symmetry.
    There are ____ of them.

Problem
Hint
3) Treena is making cone hats for the play out of felt. She has a circle of felt with diameter 2 ft and is planning on making radial cuts to get slices of the circle. She will sew or use fabric glue to join the two radial edges together. The circumference of the opening needs to be 21 inches to fit the actors' heads. How many hats can she make from one circle?
  1. Determine the circle's diameter in inches and compute the circumference:


    Circumference = ____ inches.
  2. Divide this circumference by the circumference of one hat and round down to the nearest number of hats = _____ hats.
4) The length of a rectangle is increased to 2 times its original size and its width is increased to 3 times its original size. If the area of the new rectangle is equal to 1,800 square units, how many square units is the area of the original rectangle? Use L for the original rectangle's length and W for it's width.
Write the equation for the larger rectangle's area and solve for LW



    LW = _______________
5) Initially the rectangular prism on the left was full of water. Then water was poured into the right cylindrical container so that the heights of water in both containers are equal. Find the height (h) of water in both containers. Round your answer to the nearest tenth of a centimeter.(Drawing not to scale.) Use Vp as the volume of the air in the prism after the water is poured into the cylinder.
Use Vc as the volume of the cylinder up to height h
  1. The equation for the volume of water removed from the prism is:


        Vp =
  2. The volume of water in the cylinder is (in terms of h):


        Vc = ____________
  3. Set them equal to each other and solve for h:


        h = ____