Question

How can a string of length $l$ be made into a rectangle so as to maximize the area of the rectangle?

• A. $\frac{l^2}{16}$
• B. $\frac{l^2}{8}$
• C. $\frac{l^2}{4}$
• D. $\frac{l^2}{2}$

Answer 1

The correct option is A

$\frac{l^2}{16}$

As per the question, $2(b+x)=l$
$\Rightarrow b=\frac{l}{2}-x$
Now, area, $A=xb=x(\frac{l}{2}-x)=\frac{lx}{2}-x^2$
For A to be maximum, $\frac{dA}{dx}=0\Rightarrow x=\frac{l}{4}andb=\frac{l}{4}$
Hence, $(Area{)}_{max}=\frac{l}{4}\times \frac{l}{4}=\frac{l^2}{16}$