Question

Let the dimensions of the rectangular plot of the greatest area be length$=l$ and breadth$=b$ which can be laid out within a triangle of base $36ft$. & altitude $12ft$ . Assume that one side of the rectangle lies on the base of the triangle. Find $\frac{l}{b}$ ?

Answer 1

$\Delta APQ\sim \Delta ABC$
$\frac{12-y}{12}=\frac{x}{36}$
$3(12-y)=x$,
Thus area of rectangle is $A=xy=3(12-y)y=3(12y-y^2)$
Now for area to be maximum, $\frac{dA}{dy}=0$
$\Rightarrow 3(12-2y)=0\Rightarrow y=6\Rightarrow x=18$
Hence $\frac{l}{b}$ is $3$.