How do you find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 8 cm if two sides of the rectangle lie along the legs?
Its area is
The largest possible rectangle must have a vertex that touches the hypotenuse of the triangle at one point.
Let us put the right angle of the triangle at the point
We can represent points on the hypotenuse parametrically as:
Then the area of the rectangle with vertices:
#(0, 0)#, #(8t, 0)#, #(8t, 3(1-t))#, #(0, 3(1-t))#is:
#f(t) = 8t*3(1-t) = 24t(1-t) = 24(t-t^2) = 24(1/4-(t-1/2)^2)#
This takes its maximum value when