Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 7 Measurement
Problem
Solution
1) Lexi looks at her orange and wonders what percent of the orange by volume is fruit that she can eat. She measures the circumference to be 25 cm. Then she begins to peel the orange and finds the peel to be 8 mm thick. What percent to the nearest whole percent of the orange is edible fruit?
The volume of a sphere is given by (4/3) r3 where r is the radius. 		Careful! There is a change of units here!
  1. The radius of the orange is 25/2 = 3.98 cm.
  2. The volume of the orange with the peel is
    (4/3) 3.983
  3. The volume of the orange without the peel is
    (4/3) (3.98 - .8)3
  4. The ratio of these 2 expressions is:
    (4/3) (3.18)3 = 32.16 = 0.51 = 51%
    (4/3) (3.98)3    63.04
2) The Telinos are tiling their bathroom. Part of the design will be a zigzag band all around the room. To create this they will need to cut white triangles and grey parallelograms from rectangle tiles. The rectangles have dimensions 4 cm by 12 cm as labeled. The triangles have height 7 cm. What is the perimeter of a parallelogram tile in centimeters? Round to the nearest whole centimeter.
See the figure to the left for the missing dimensions.
  1. The missing dimension is the diagonal which is the hypotenuse of the triangle.
  2. Using the pythagorean theorem:
    c = √ (a 2 + b 2) 
  3. Plugging in our values:
    c = √ (4 2 + 7 2) 
  4. c = 8.06 cm.
  5. So the perimeter of the parallelogram is
    2(5) + 2(8.06) = 26.12 cm = 26 cm

Problem
Solution
3) At Mario's Pizza they serve a pizza lunch buffet. The pieces are cut from 8 inch diameter pizzas and 12 inch diameter pizzas. The 8 inch pizzas are cut to have 8 congruent slices. The 12 inch pizzas are cut so that the pieces have the same area as the pieces from the 8 inch pizza. What is the angle at the tip of a slice from the 12 inch pizza?
  1. The area of one of the slices from the 8 inch pizza = ( 42)/8 sq. in.= 2 sq. in.
  2. The area of the 12-inch pizza is:
    62 = 36 sq. in.
  3. Dividing the area of the 12-inch pizza by the area of one slice of the 8-inch pizza =
    36 = 18
     2
  4. This means that the large pizza must be cut into 18 slices, so the angle at the tip is 360/18 = 20 degrees
4) Shauna enters a bike race. She checks out the course one weekend and it takes her a full hour to ride the course. If she rides at an average rate of 18 mph for the first 20 minutes, speeds up to 25 mph for the next 10 minutes, slows to 12 mph for the next 20 minutes and finishes with a speed of 28 mph, how long is the course? Express your answer to the nearest tenth of a mile. Compute the distances for each leg and add:
  1. 18 mi x 20 min = 6 miles
    60 min
  2. 25 mi x 10 min = 41⁄6 miles
    60 min
  3. 12 mi x 20 min = 4 miles
    60 min
  4. The remaining time out of the hour is
    60 - 20 - 10 - 20 = 10 minutes
  5. 28 mi x 10 min = 42⁄3 miles
    60 min
  6. The course is 6 +41⁄6 + 4 + 42⁄3 miles =
    185⁄6 miles = 18.8 miles
5) Mr Green is building a raised bed planter box frame 14 ft. long, 10 ft. wide, and 3 ft. high as shown. The top is 2 feet across from inner edge to outer edge so that Mr. Green can sit down comfortably. Mr. Green plans to treat all the surfaces, including the bottom, with a weather resistant chemical. What is the surface area of the entire frame so that he will know how much chemical to purchase?
 There are many ways to cut up this planter box to sum up a series of rectangles, but the figure to the left is one.
  1. Top and bottom
    2 (14x2 + 14x2 + 6x2 + 6x2) = 160 sq. ft.
  2. Outside:
    10x3 + 10x3 + 14x3 + 14x3 = 144 sq. ft.
  3. Inside:
    10x3 + 10x3 + 6x3 + 6x3 = 96 sq. ft
  4. Total = 160 + 144 + 96 = 400 sq. ft.