Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 8 Measurement
Problem
Solution
1) Georgia buys a waffle cone and gets a single scoop of ice cream. The scoop of ice cream is a ball with a diameter of 3 inches. The cone has a height of 6 inches and a diameter of 3 inches. Tony buys the identical cone but his cone is completely filled to the top with ice cream and sealed in with a waffle cookie across the top. What is the ratio of the volume of Georgia's scoop of ice cream to the volume of Tony's cone of ice cream? Express your ratio as a fraction in lowest terms.
Volume of a sphere is (4/3) r3
Volume of a cone is (1/3) r2 h
Use = 3.14 		Ratio =
(4/3) r3 = 4 r /h = 4 x.1.5 / 6 = 1/1
(1/3) r2h

2) Voyager 1 was launched in 1977 to go out and collect data on our solar system and beyond. It takes light 17 hours to travel to earth from Voyager's location. A light year is the distance light travels in a year and is 9.46x1015 meters. How far away from earth is Voyager in meters?
Assume a year is 365 days and each day is 24 hours long. 		17 hours x 9.46x1015 m = 1.836x1013 meters
365 x 24 hours

Problem
Solution
3) HD TVs screens have a length to width ratio of 16:9 and are described by their diagonal length. Kevin just purchased an HD TV with diagonal length 92 inches. He wants to purchase an entertainment center to go with it. There is an added 1 inch border all around the screen. What are the minimum length and width dimensions for the space inside the entertainment center to fit the TV? Round your answers to the nearest inch that will fit the TV.
  1. The diagonal length of a TV that is 16x9 inches is, by the pythagorean theorem

    D = √ 16 2 + 9 2 
  2. D = √ 337
  3. D = 18.36 inches
  4. Applying the scale factor of 92/18.36 to the length and width:
    L = 16 x 92 / 18.36 = 80.17 inches
  5. The width W =99 x 92 / 18.36 = 45.1 inches
  6. Add the 1 inch borders (2 of them) and round up:
    L = 83 inches
    W = 48 inches
4) Sara asks Ryan to figure out the total area of the sail for the 12 foot sailboat she is making. She draws a scaled picture of the sail she wants to make from a photograph and writes in the lengths from the picture, see drawing. Ryan sees it is an isosceles triangle. If she wants the bottom of the sail for her boat to be 13.5 feet long, what will the area be?
 Set up the ratio of the scale drawing's base BC to the real sail's base, 13.5 feet, and apply to the height to determine the actual sail's area.
  1. The ratio of the actual sail's width to the scale picture's base (BC) is:
    13.5/4.5 = 3
  2. The actual sail's height is 4.15 x 3 = 12.45 feet
  3. Area = (13.5 x 12.45)/2 = 84.0375 sq. ft.
Note: The fact that Ryan sees it as an isosceles triangle is irrelevant to the problem. Also, once you know the ratio of the lengths (3) you can take the scale model's area (4.15x4.5/2) and multiply it by the square of the ratio (9) to get the area:
(4.15x4.5/2)x9 = 84.0375.
5) On April 2, 2013 Mrs Green planted two different bushes three feet apart, stem to stem. They both have a diameter of 1 foot. The diameter of the first bush grows at a rate of 6 inches per year. The diameter of the second bush grows at a rate of 4 inches per year. Assume the growth rates are constant throughout the year. In what year and month will the two bushes touch and begin to form a hedge?
  1. If the bushes are planted 3 feet (36 inches) apart and they both have a diameter of 1 foot, then they are separated by 36 - 2x6 = 24 inches.
  2. The gap between them narrows at a rate of (6/2 + 4/2) = 5 inches per year.
  3. It will take 24/5 = 4.8 years = 4 years 9 months to close them.
  4. From April, 2013 this is January 2018