Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 8 Measurement
Problem
Solution
1) A rectangle has length 4 inches and width 3 inches. An isosceles triangle is formed by cutting the rectangle along its diagonal and joining the two triangles formed along the long side of the rectangle, see diagram. What is the perimeter of the new triangle?
  1. The length of the diagonal (by the pythagorean theorem) is 5 in. (We have a 3-4-5 triangle).
  2. The perimeter of the new triangle is 5x2 + 6 = 16 inches
2) What is the area of a regular octagon with side length 1 cm (see picture)? You may leave square roots in your answer but simplify as much as possible or round to the nearest thousandth.
Solution(s)
Method 1 (easier method): Compute area of isosceles triangle BDE, add 4 of them plus the rectangles:
  1. Determine the length of segment S, using the pythagorean theorem:
    1 = √ S 2 + S 2)  = S√2
    S = 1 / √2 = √2 / 2
  2. Compute area of triangle BDE = (√2 / 2)2/ 2 = .25
  3. 4 of these triangles = 1
  4. Determine area of rectangle CDEH = √2 / 2
  5. 2 of these = √2
  6. The length of line AB = 1 + 2/ √2 = 1 + √2
  7. Total all parts:
    1 + √2 + 1 + √2 =
    2 + 2√2
Method 2: Cut into trapezoids and add up the parts.
  1. Determine the length of segment S (see Method 1)
    = √2 / 2
  2. The length of line AB = 1 + √2
  3. The area of trapezoid ABDC is
    (AB + 1) S/2
  4. 2 trapezoids =
    (AB + 1) S = AB x S + S
  5. 2 trapezoids + rectangle (the whole octagon)=
    AB x S + S + AB =
    AB (S + 1) + S
  6. Substituting our values for AB and S:
    (1 + √2) (√2/2 + 1) + √2/2 =
    ( √2/2 + 1 + 1 + √2) + √2/2 =
    √2 + 2 + √2 =
    2 + 2√2 sq. cm.




Method 3:
(possibly cheating!):
Use the equation for the area of a regular octagon:
A = 2 (1 + √ 2) s2
s = 1 cm,
so the area is:
2 + 2√2 sq. cm.

Problem
Hint
3) 800 liters of water are poured into a circular inflatable swimming pool. If the diameter of the pool is 160 centimeter how deep is the water to the nearest centimeter?
1 liter is the same as 1000 cubic centimeters of water.
You may use = 3.14

  1. The volume of the pool is 800 x 1000 = 800000 cu. cm.
  2. Volume of a cylinder = r2 h
    800,000 = 802 h
  3. h = 800,000 / ( x 6400)
  4. h = 39.8 cm = 40 cm.
4) An odd little man lives by himself in an isolated house that no one ever visits. He loves money. He has covered one 120 square foot wall with $10 bills with no gaps and no overlaps. He decides instead he wants to enjoy the sight of the $10 bills in many rooms and will make a continuous strip of them along the hallways and in and out of rooms. He can place the $10 bills with no gaps or overlaps so that the strip is 1/2 foot wide or 5/24 foot wide. How much longer would the 5/24 foot wide strip be than the 1/2 foot wide strip?
  1. Length of the 1⁄2 foot wide strip =
    W = 120 / .5 = 240 feet.
  2. Length of the 5⁄24 wide strip =
    W = 120 / 5⁄24 = 120 x 24⁄5 = 576 feet.
  3. 576 - 240 = 336 ft.
5) A wooden shipping crate is measured for shipping. Its length is 8 feet and its girth (going completely once around in the direction perpendicular to the length) is 24 feet. The surface area of the lid of the crate is 40 square feet. The dimensions are whole numbers of feet. What is the volume of the crate in cubic feet?
  1. If the top area is 40 sq. ft. then the width of the lid is 40/8 = 5 ft.
  2. Using the girth of 24 feet, the other dimension must be (24 - 10)/2 = 7 feet.
  3. Therefore the volume is 8 x 5 x 7 =
    280 cu. ft.
    See the figure to the left for these dimensions.