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.
