Homework Help Forum: Calculus

Posted by math on Monday, October 26, 2009 at 6:26pm.

What is the area of the largest rectangle that can be placed in a 5-12-13 right triangle (as shown)?

• Calculus - MathMate, Monday, October 26, 2009 at 7:32pm

  Let ABC be the vertices of the triangle, right-angled at B, AB=5, BC=12 (vertical side), AC=13. Draw a rectangle BDEF, where D is on AB, E is on AC and F is on BC. Denote x=DE= height of rectangle Width of rectangle = DB = 5-(5x/12) Area of rectangle, A(x)=x(5-(5x/12))=5x-5x²/12 A'(x) = 5-10x/12 For A(x) to be maximum, A'(x) = 0 = 5-10x/12 x=6, 5-5(6)/12 = 2.5 The maximum area is 6*2.5=15 Note that the aspect ratio of the rectangle is the same as that of the right sides of the triangle.

  
  

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