As part of the Community Service Action Team’s mission to offer diverse service opportunities for employees, engineers from EDG's Metairie office participated in the regional MATHCOUNTS competition that was held at UNO on Saturday, February 6th. Several area engineering firms sponsor and/or volunteered at this event, which encourages enthusiasm and achievement in middle school mathematics. This year, at over 12 area schools participated.
Check your MATHCOUNTS skill –
Questions:
- Jean made a New Year’s resolution to get in shape. She decides to run for 30 minutes on Tuesdays, Thursdays, Saturdays and Sundays. If Jean plans to run at an average speed of 6 mph, how many miles will she run during the month of February 2010?
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- After some consideration, Jean decides that it might be better to run at an average speed of 5.0 miles per hour on her first day of running in February and then increase her average speed by 0.1 miles per hour each day she runs. How many fewer miles will she run than if she were to run at an average speed of 6.0 mph each time she runs?
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- By following the new plan, by what percent will Jean have increased her average speed from the first running day in February to the last running day in February?
Answers:
- 48 Miles
- 2 Miles fewer
- 30% increase
Solutions:
1)
There are 4 Tuesdays, 4 Thursdays, 4 Saturdays and 4 Sundays in February, which means that she will run for (0.5 hours)(4)(4) = 8 hours during February. Thus, Jean will run 8(6) = 48 miles in February.
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2)
(5.0)(0.5) = 2.5 miles
(5.1)(0.5) = 2.55 miles
(5.2)(0.5) = 2.6 miles
(5.3)(0.5) = 2.65 miles
We quickly see that this forms an arithmetic sequence, in which, each time she increases her speed by 0.1 mph, her distance increases by 0.05 miles. To find the sum of the number of terms in an arithmetic sequence, you find the sum of the first term and the last term, multiply it by the number of terms, and then divide by 2. So let’s find the last term (the number of miles she would run on her last running day in February).
a1 + (d (n – 1)) = an
2.5 + (0.05 (16 – 1)) = 3.25. Thus, the total number of miles Jean would run is ((2.5 + 3.25)(16))/2 = 46 miles
That is 48 – 46 = 2 miles fewer than if she ran at an average rate of 6 mph each time she ran in February.
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3)
If Jean starts February at an average speed of 5.0 miles per hour and increases her speed by 0.1 miles per hour each time she runs, the last day she runs in February she will run at an average speed of 5.0 + (0.1(16 – 1)) = 6.5 mph. That is an increase of 6.5 – 5.0 = 1.5 mph, which is a percent increase of (1.5/5.0)(100) = 30%.
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