Garden paving slabs measure 40 cm by 20 cm - Leaving Cert Mathematics - Question (b) - 2012

Given that the perimeter is 64 metres, we can convert this to centimetres:

$P = 6400 ext{ cm}$

Setting the perimeter equation equal to 6400 gives:

$80x + 40y = 6400$

Solving for y, we get:

$40y = 6400 - 80x$
$y = 160 - 2x$

The area (A) can be represented as:

$A = 40x imes 20y = 800xy$

Substituting for y from the earlier work, we have:

$A = 800x(160 - 2x) = 128000x - 1600x^2$

To find the maximum area, we take the derivative and set it to zero:

$\frac{dA}{dx} = 128000 - 3200x = 0$

Solving for x gives:

$3200x = 128000 \Rightarrow x = 40$

Using the value of x found, we substitute back to find y:

$y = 160 - 2(40) = 80$

Now, the number of slabs is given by:

$\text{Number of slabs} = xy = 40 \times 80 = 3200$

Therefore, a total of 3200 slabs are needed to pave the maximum area.