Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2015 Grade 7 Geometry
Problem
Hint
1) How many squares are in this figure?
 The key to this kind of problem is identifying each size square and then counting them.
  1. The smallest squares: ____
  2. The squares that are 2 of the smallest squares on a side: ___
  3. The squares that are 3 of the smallest squares on a side: ___
  4. The squares that are 4 of the smallest squares on a side: ___
  5. The squares that are 5 of the smallest squares on a side: ___
  6. Total = ______

2) Jacob wants to measure the height of his house using shadows. Jacob is 5 feet 10 inches tall and casts a shadow that is 2 feet 6 inches long. His house casts a shadow that is 9 feet 6 inches long. What is the height of Jacob's house? 		Let H = house
  1. Set up the ratio of Jacob's height/Jacob's shadow and set it equal to House height/House shadow:
  2. Solve for H:


        H = ______

3) A container used for gardening measures 14 in across and 3 ft wide, capped with a half circle at each end as shown. It is 3/4 ft deep. What is the volume of soil needed to fill it?
(Express your answer to the nearest tenth of a cubic feet.)
 The area of the container is the sum of a __ x ____ inch rectangle and a circle of radius ___ inches. (each of the 2 half circles makes a whole one)
  1. Area of the rectangle:


    = _______
  2. Area of the circle


    = __________
  3. The total area = ____________
  4. The volume is ____________ cu. ft.

Problem
Solution
4) A rectangular wooden block measuring 8 cm by 14 cm by 18 cm is painted green all around and then cut into 2 cm cubes. What is the ratio of cubes that are painted on one or more sides to cubes with none of the faces painted? Expressed your answer in the lowest term.
  1. Count the total number of 2 cm cubes = ____ cubes
  2. The unpainted cubes are the interior cubes (subtract 2 from each dimension in cubes):
        = _____ cubes
  3. The cubes with some paint on them is ______
  4. The ratio of the painted to unpainted cubes is
    _________

5) XYZ is an equilateral triangle, 12 units to a side. X, Y, and Z are also the center points of three equal circles that intersect without overlapping. In the figure below, what is the area of the shaded section to the nearest tenth of a unit?
(Use 3.14 for pi)
  1. The area of an equilateral triangle is s2(√ 3) / 4
    where s is the length of the side
  2. For our triangle, this is ____________
  3. The radius of one of the circles is ___ units (half of the triangle side length), so the area of one of the circles is
    ______________________
  4. So the triangle angles are _____ of the circle, so the portion of each circle that is inside the triangle is ______ of the circle = ____________
  5. Since there are 3 of them inside the triangle the total area is _______________
  6. Subtracting this from the area of the triangle:
    Shaded area = ___________