The greatest area of the rectangular plot which can be laid out within a triangle of base 36 feet and altitude 12 feet equals ( assume that one side of the rectangle lies on base of triangle).

draw a triangle with base AB= 36 and altitude CD= 12 such that the foot of this altitude on base (of length 36) is at a distance x from the right. Or DB=x. so DA= 36 – x. now assume a rectangle of height y. let it meet AC at P and BC at Q.

also let it intersect CD at R.

now, CR/PR= CD/AD or (12 – y)/PR= 12/(36 – x) similarly CR/RQ= CD/DB or (12 – y)/RQ= 12/x

now, area of rect= y(PR+RQ)= y((12 – y)(36 – x)/12 + (12 – y)x/12)

= [y(12 – y)/12](36 – x + x)

A= 3y(12 – y).

for max area, dA/dy=0 or 12 – 2y=0 or y=6

so, Amax= 3*6*6

= 108 feet^2