How do you find the right triangle of maximum area if the sum of the lengths of the legs is 3?

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Answer 1 · 2015-06-03T18:54:11

We know that #x+y=3# and that the area #A = (x*y)/2# because of the right angle.

Let's use substitution and replace #y# in the equation of the area :

#x+y=3<=>y=3-x#

#A = (x*(3-x))/2 = (-x^2+3x)/2#

The maximum area can be found by studying the sign of the derivative of the area :

#A'(x) = (-2x+3)/2#

The derivative #= 0# when #-2x+3=0<=>x=3/2#.
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Therefore, the area is maximal when #x=3/2# :

#A=(-(3/2)^2+3*(3/2))/2 = (-(9/4)+(18/4))/2 = 9/8#.