Maximum Area of a Triangle with Sides of

9, 40x, and 41x

 

Given a triangle with one side of length 9 units and the ratio of the other two sides is 40/41.

 

Provide an intuitive argument as to why this triangle would have a maximum area.  Note that the figure here is not to scale and may be misleading.  As x increases  and approaches 9, the triangle becomes obtuse. We must have  0 < x < 9.

 

Find the Maximum area.

     Hint:   Use Heron's Formula to set up a function for the area of a 9 - 40x - 41x triangle.

     Exploration:   Try a spreadsheet for  1 ≤ x ≤ 9 See Graph

x a b c s  Area
1 9 40 41 45 180
2 9 80 82 85.5 354.84
3 9 120 123 126 515.13
4 9 160 164 166.5 652.79
5 9 200 205 207 757.50
6 9 240 246 247.5 814.91
7 9 280 287 288 801.76
8 9 320 328 328.5 667.88
9 9 369 360 369 0

 

           Use the AM-GM Inequality to show the maximum area.

 

GSP file?


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