Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 8 Measurement
Problem
Solution
1) Mikki's Ice Cream is planning on changing their ice cream packaging from rectangular cartons to cylindrical containers. Their cartons measure 7 in x 5 in x 4.5 in. To keep the amount of packaging about the same they would like the surface area of the carton to be the same as the surface area of the new cylindrical containers (top + bottom + side). The height to diameter ratio should be 1.5 to 1. What are the radius and height of the cylinder to the nearest tenth of an inch?
Use = 3.14 or provide an exact answer.
  1. The surface area of the rectangular cartons is 2x7x5 + 2x7x4.5 + 2x5x4.5 = 178 sq. in.
  2. Using h for the height of the cylinder and d for the diameter, the surface area of the cylinder =
    hd + 2 (d⁄2)2
  3. Now, if the height is 1.5 times the diameter (= 3d/2), then
    (3d2/2) + 2 (d⁄2)2
  4. Simplify and set equal to the rectangular box surface area:
    (3d2/2) + 2 d2⁄4 = 178 sq. in.
    (3d2/2) + d2⁄2 = 178 sq. in.
    2d2 = 178 sq. in.
    d2 = 89 /
    r = 5.322 / 2 = 2.7 inches
    h = 3d/2 = 3x5.322 / 2 = 8 inches
2) A turtle is racing a hare. The race course is 60 feet long. The turtle trundles along at a rate of 40 ft per hour without resting. The hare hops along at a rate of 60 ft per hour but rests for 15 minutes every 25 feet.
a. How does the race end?
b. What is the distance between the winner and loser?
  1. The turtle takes 60 feet / (40 ft/hour) = 1.5 hours
  2. The hare hops 25 feet and takes 25 / 60 ft/hour = 25 minutes
  3. So the hare takes:
    1. 25 minutes to go 25 feet:
      Total distance = 25 feet total time = 25 minutes
    2. Rests for 15 minutes:
      Total distance = 25 feet total time = 40 minutes
    3. Hops for another 25 feet:
      Total distance = 50 feet total time = 65 minutes
    4. Rests for another 15 minutes:
      Total distance = 50 feet total time = 80 minutes
    5. Hops for the remaining 10 feet taking 10 minutes:
      Total distance = 60 feet; total time = 90 minutes
  4. They cover the same distance in the same time (90 minutes).

Problem
Solution
3) A goose is being tracked on its migration south by a tiny device. Its flight path for two legs of its journey is shown on the grid which is centered at Walla Walla, numbers are in miles. How far did the goose fly over the two legs of the journey? Express your answer to the nearest mile. We must use the pythagorean theorem to determine the flight path lengths.
  1. First leg: D1 = √ 50 2 + 120 2)  =√16900 =
    130 miles
  2. Second leg: D2 = √ 70 2 + 40 2)  =√6500 = 80.62 miles
  3. Total distance = 130 + 80.62 =
    210.62 = 211 miles
4) The diameter of Earth is 1.28 x 104 km and the diameter of Jupiter is 1.43 x 105 km. Assume both planets are spheres. How many times larger to the nearest whole number is Jupiter's surface area than Earth's surface area? Surface area of a sphere: 4 r2 , where r is the radius.
  1. Jupiter's surface area = 4 (0.715x105)2 sq. km.
  2. Earth's surface area = 4 (0.64x104) 2 sq. km.
  3. Their ratio = (0.7152x 1010) / (0.642x108) = 1.2468 x 102 = 125 times larger

5) Dorey goes to Italy and Greece every year for about half the year. She brings back clothing to sell. At the beginning of her trip the dollar euro exchange rate is $1.00 = € 0.66. She buys 10 silk blouses at € 35.00 a piece. When she returns to the US the exchange rate is $1.00 = € 0.78. Does the cost in dollars for one silk blouse go up or down from the beginning to the end of her trip and by how much?.
  1. Cost of blouses in dollars = 35/0.66 = $53.03/blouse
  2. Converting back to dollars at the end of the trip =
    35/0.78 = $44.87
  3. The cost in dollars went down by 53.03 - 44.87 = $8.16