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

Answer 1

Place the right angle at the origin and let the 3 cm length be along the x axis and the 4 cm length along the y axis.
Let x be the distance of one corner from the origin along the x axis.
The hypotenuse has the equation
y = 4 - (4/3)x
For the rectangle thus formed,
A = x*y = 4x - (4/3)x^2
This has a maximum when
dA/dx = 4 - (8/3) x = 0
x = (3/8)*4 = 1.5
y = 4 - (4/3)(3/2) = 2
A = 3 cm^2