Wow W.

asked • 01/11/15

What is the median of the first 2015 even integers?

I have this math problem given by my teacher. I have no idea how to do it. This is the full question below:
 
It’s 2015! Let’s have a little fun with this number…

What is the median of the first 2015 even integers?

How many minutes after noon is it when a 24-hour digital clock displays the correct time as 20:15?

If the perimeter of regular pentagon ABCDE is 2015 inches, what is the area of triangle BCE, in square inches? Express your answer as a decimal to the nearest hundredth

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Douglas W. answered • 01/11/15

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Douglas W. High School Algebra, Geometry, Reading, Vocab, SAT

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The median is the middle number (integer) or the average of the middle two numbers in the series. 
 
In the case of 2015, the middle two numbers would be 1007 and 1008.  Their average would then be 1007.5
 
Doug W.
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Mark M.

The question was about the 2015 even integers.
So the median would be between the 1007th and 1008th even integer, i.e., between 2014 and 2016 which is 2015!
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01/11/15

Douglas W.

Right Mark.  Just glanced at the thing early in the AM. 
 
The median of first 2015 even integers would be a series of even integers starting with 0 and ending with 4030.
 
The median would be the average of 2014 and the next integer in the list: 2016, which is 2015. 
 
All this assumes you accept that 0 is an even integer: see http://mathforum.org/library/drmath/view/57188.html
 
I also now noticed the other questions in Wow's list. The 24 hour clock question, appears to be the difference between 12:00 and 20:15, or 60 x 8 (hours) + 15, which is 495 minutes.
 
The pentagon question requires illustrations and mathematical symbols that I don't know how to reproduce in this simple message format (sigh!) 
 
Drawing the pentagon, you see that the triangle formed by BCE is one of three triangles produced by connecting those points.  The other two triangles are isosceles triangles with the two sides formed by the pentagon's sides.  We know the length of these sides, since they are the pentagon's circumference divided by 5 (2015/5) or 430.  We also know the included angle of the isosceles triangle, since each vertex of the pentagon is 180(5 sides-2), or 108 degrees. 
 
 
The formula for the area of an isosceles triangle given the the two legs and the included angle is s2 sin(included angle/2) times cos(included angle/2) or 4302(sin(540)cos(540)
 
Now we know the area of each of the other two triangles that the pentagon has been divided into.  Subtracting them from the area of the pentagon gives the area of the triangle we drew.
 
So how do we get the area of the pentagon?
 
the area of a regular polygon is the square of the length of a side times the number of sides all of this over 4 times the tangent of (180/number of sides).  So the area of this pentagon is 430(52)/4(tan(180/5 degrees)) or 10750/4(tan360) or 10750/4(0.726542528)
 
 
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01/11/15

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