Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2016 Grade 8 Geometry
Problem
Solution
1) The circumference of a major league baseball is 235 mm. If the volume of a sphere is given by the formula V = 4/3(pi)r3, where r is the radius, what is the volume of the baseball to the nearest cubic mm? (Use 3.14 for pi)
  1. First, find the radius of the baseball:
        r = C / 2 = 235 / 6.28 = 37.42 mm.
  2. V = 4⁄3 (37.423) = 52399.2 x 3.14 x 1.3333 = 219,377 cu. mm.

2) Write the letters M, T, and H in the correct orientation on figure 1 so that when the net is folded into a box, it will spell MATH vertically as shown in Figure 2.
  1. If you fold the 2 spaces to the left of the "A" in, then the lower of the 2 spaces holds the "T", but it must be sideways
  2. If you fold the 2 spaces to the right of the "A" in, then the upper of the 2 spaces holds the "M", again, sideways.
  3. The two empty spaces to the left and right of the "A" remain empty because they make the left and right sides of the folded cube.
  4. The "H" is in the upper right space and it, too, is sideways, resulting in the figure you see to the right.
3) Farmer Cobb is trying to construct a pen with 8 sections of fence that are each 18 feet long. The sections are rigid and cannot be bent. What is the greatest area he could encompass with the pen? (Round your answer to the nearest tenth of a square foot)
  1. Consider that if you had 1000 sections how would you make the largest area with them? You would approximate a circle with them!
  2. The figure that 8 sections can make that most closely approximates a circle is an octagon
  3. The area of an octagon is
        A = 2 (1 + √ 2) s2 where s is the side length.
  4. For our octagon with a side length of 18 feet:
    A = 2 (1 + √ 2) 182 =
    A = 2 (1 + √ 2) 324 =
    4.8284 x 324 = 1564.4 sq. ft.

Problem
Solution
4) On an analog clock, you observe that the angle between the hour and the minute hands is 90 degrees at 3 pm. What will the angle between the hands be 35 minutes later? Express the answer to the nearest tenth of a degree. (Both hands move at different constant rates)
  1. The entire clock face has 12 hours.
  2. You must figure out how much the hour hand moves and how much the minute hand moves.
  3. The minute hand moves:
    35/60 x 360 degrees = 7⁄12 x 360 = 210 degrees.
  4. The hour hand moves:
    35/(60x12) x 360 = 17.5 degrees.
    The hour hand moves forward 17.5 degrees from it's initial position at 90 degrees = 107.5 degrees.
  5. The angle between the hour and the minute hand is 210 - 107.5 = 102.5 degrees
5) What is the area of the shaded region of the square, with an equilateral triangle inside, with a side length of 5 units? (Express your answer to the nearest hundredths of a square unit)
 Method 1: Compute the area of the triangle and subtract:
  1. Using the pythagorean theorem, the height of the equilateral triangle (h) is
    So,  h = √ s 2 - (s/2) 2 
  2. h = √ s 2 - s2/4  
  3. h = √ 3/4(s 2) 
  4. h = s √ 3/4   = .866 s
  5. The area of the equilateral triangle is:
    .866 s x s/2 = .433 s2 = .433 x 25 = 10.825 sq. units
  6. The area of the shaded region =
    25 - 10.825 = 14.175 sq. units
Method 2: Use the formula for the area of an equilateral triangle
  1. The formula for the area of an equilateral triangle is:
    s2(√ 3  ) / 4
  2. The shaded area is s2 - s2(√ 3  ) / 4 =
    s2(1 - (√ 3  ) / 4) =
    25(1 - .433) = 25 x .567 = 14.175 units