A piece of wire 6 metres long is cut into two pieces - NSC Mathematics - Question 9 - 2017 - Paper 1

The remaining length of the wire after forming the square is:

$6 - x$

The length of the rectangle BEFC can be calculated as:

$2l + x + \left(6 - x\right) = 6$

Substituting $l$ into the equation gives:

$2\left(\frac{x}{4}\right) + x + \left(6 - x\right) = 6$

The area $A$ of the U-shaped rectangle can be given by:

$A = l \times \left(\frac{x}{4} + 6 - x\right) = \left(\frac{x}{4}\right) \left(24 - 5x\right)$

So,

$A = \frac{x(24 - 5x)}{4}$

To find the maximum area, we need to take the derivative of the area with respect to $x$ and set it to zero:

$\frac{dA}{dx} = 0 \Rightarrow \frac{24 - 6x}{4} = 0$

Solving for $x$ gives:

$6x = 24 \Rightarrow x = 4$