Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2015 Grade 7 Geometry
Problem
Solution
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: 13
  2. The squares that are 2 of the smallest squares on a side: 8 of these
  3. The squares that are 3 of the smallest squares on a side: 5 of these
  4. The squares that are 4 of the smallest squares on a side: none of these
  5. The squares that are 5 of the smallest squares on a side: 1 of these
  6. Total = 13 + 8 + 5 + 1 = 27
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:
        70 in = H
        30 in   114 in
  2. Solve for H:
        H = 114 (70) / 30 = 266 in. = 22 ft 2 in
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 14x36 inch rectangle and a circle of radius 7 inches. (each of the 2 half circles makes a whole one)
  1. Area of the rectangle: 14x36 = 504 sq. in.
  2. Area of the circle = 72 = 49 = 153.86 sq. in.
  3. The total area = 504 + 153.86 = 657.86 = 657.9 sq. in. (rounded)
  4. The volume is 657.9 x 9 in. = 5921.1 cu. in.
  5. Convert to cubic feet:
        5921.1/1728 = 3.4 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 =
        4x7x9 = 252 cubes
  2. The unpainted cubes are the interior cubes (subtract 2 from each dimension in cubes):
        2x5x7 = 70 cubes
  3. The cubes with some paint on them is 252-70 = 182
  4. The ratio of the painted to unpainted cubes is
    182/70 = 91/35 = 13/5
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)

 Method 1: Use the equation for the area of an equilateral triangle:
s2(√ 3) / 4
where s is the length of the side
For our triangle, this is (144x1.732)/4 = 62.352 sq. units.

Method 2: Derive the equation for the area of an equilateral triangle (when you don't know it):
See the figure to the left.
  1. The height of the triangle is
    (by the pythagorean theorem) =
    h = √ 12 2 - 6 2  = √108
  2. The area of 1 triangle = 6 √108 = 62.352 sq. units.
Now that we have the area of 1 equilateral triangle, the solutions to both methods are the same:
  • The radius of one of the circles is 6 units (half of the triangle side length), so the area of one of the circles is     36
  • So the triangle angles are 1⁄6 of the circle, so the portion of each circle that is inside the triangle is 1⁄6 of the circle = 6
  • Since there are 3 of them inside the triangle the total area is 18
  • Subtracting this from the area of the triangle:
    Shaded area = 62.352 - 18x3.14 =
    5.832 sq. units. = 5.8 sq. units rounded