Question 1138788
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You don't even have to get into trigonometry with the sine of an angle to solve this problem.<br>
Given the lengths of two sides of a triangle, the maximum area of a triangle with those two side lengths is when those two sides form a right angle.<br>
That should be easy to see by considering one of the two given side lengths as the base.  The area of the triangle is one-half base times height; and clearly the maximum height of the triangle is when the second given side is at right angles to the first.<br>
And, to answer the question, the maximum area of a right triangle is one-half the product of the two legs; so the maximum possible area with two legs of length 10m is (1/2)(10m)(10m) = 50m^2.