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posted by  ho_ney777 on 6/28/2008 3:20:57 PM  |  status: Live  

Calculus optimization problem

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Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3cm and 4cm if two sides of the rectangle lie along the legs.

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posted by Maxp on 6/28/2008 3:45:43 PM  |  status: Live
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From the Diagram the Area is: A =xy
 
since the corner has to lie on the the triangle, we must find the equationof the line that forms the hypotenuese.
 
m = (3-0)/(0-4)
   = -3/4
 
y = mx+b, the y-intecept = 3
 
so y = -3/4x +3
 
∴ A = x(-3/4x+3)
          =-3x2/4 +3x
 
taking the derivative and setting it to 0
 
we have
0 =-3/4(2x)+3
x=2   and so y= -3/4(2)+3
                      =1.5
So Max Area is:
A = 3 cm2
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