Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 8 Measurement
Problem
Solution
1) Nadia wants to run a 5 mile marathon in June. In early March, she could only run for 15 minutes and go a distance of one mile in that time. By early April her speed had gone up by 30%. If every month she improves by 30% over the previous month in speed, how long will it take her to run the 5 mile marathon in June? Round to the nearest hundredth of a second.
  1. Nadia's starting speed is 1 mile ÷ 1⁄4 hour = 4 miles/hour.
  2. Her speed increases by 30% per month for 3 months (until June).
  3. Final speed = 4(1.3)3 = 4x2.197 = 8.788 miles/hour
  4. Her time for the 5 mile run =
    5 miles/8.788 miles/hour = 0.5689 hour =
    34 minutes 8.25 seconds
2) Sharon takes a trip and travels at the speed limit on the different roads within seconds of getting on the road. Her trip in speed versus time is shown on the graph. What is her average speed? Express your answer to the nearest mile per hour. Compute the total distance Sharon drives and divide by the time:
  • 10 minutes at 25 mph = 1⁄6 x 25 = 4.167 miles
  • 20 minutes at 30 mph = 1⁄3 x 30 = 10 miles
  • 20 minutes at 60 mph = 1⁄3 x 60 = 20 miles
  • 10 minutes at 60 mph = 1⁄6 x 30 = 5 miles
  • Total distance = 39.167 miles taking a total of 60 minutes = 1 hour.
  • Average speed = 39.167 = 39 mph
3) Leslie builds a model pyramid with a square bottom. The square bottom has side length 6 centimeters. The triangular sides are all identical and have height 4 centimeters. Her class is going to make a big scale model of her pyramid. All dimensions will be increased by a scale factor of 50. What is the height of the large pyramid? Express your answer in meters. Multiply all dimensions by 50 and convert to meters:
base = 6x50 = 300 cm = 3 m.
height = 4x50 = 200 cm = 2 m.
Height = √ 2 2 - 1.5 2)  = √1.75 or 1.329 m.

Problem
Solution
4) At an obstacle course you can either go up and over a hill ringing the bell hanging on top, or you can follow the half circle path around the base of the hill to the same point. The hill has a height of 100 feet. The half circle path has a radius of 100 feet. See the schematic drawing. What is the ratio of the length of the path going up and over to the path which goes around the half circle? Give your answer as an exact answer or round to the nearest tenth using = 3.14.
  1. The distance around half the hill = 100 ft.
  2. The length of the side of the hill (by the pythagorean theorem) =
    Side = √ 100 2 + 100 2)  = √20000 or 141.4 ft.
  3. 2 of these = 282.8 ft.
  4. The ratio of the up-and-over to the go-around path is
    282.8/(100 ) = 0.90/1

5) Dray places a 2 liter bottle under a drippy faucet. It drips at a rate of 1 drip every 5 seconds. Three drops fill a teaspoon which is 5 milliliters.
One milliliter = 10-3 liters. How long before the bottle is full? Express your answer in hours, minutes, and seconds. 		Set up the string of conversion factors, making sure that they all cancel to yield seconds:
2 liters x 1 millileter x 3 drops   x   5 seconds = 6000 seconds
                10-3 liter       5 milliliters     drop

6000 seconds = 1 hour 40 minutes