Norman windows consist of a rectangle topped by a semi-circle as shown above - Leaving Cert Mathematics - Question 9 - 2019

Given the perimeter P = 12, we substitute this into our equation:

$12 = 2y + x + \pi x$

Rearranging gives:

$2y = 12 - (1 + \pi)x$

Thus,

$y = \frac{12 - (1 + \pi)x}{2}$

As for the domain of x, since the window must have positive dimensions, we find:

$x \leq \frac{12}{2 + \pi}$

This derives from the fact that both width and height must remain non-negative.

The slope of the function $y = \frac{12 - (2 + \pi)x}{2}$ can be found by differentiating y with respect to x:

$m = -\frac{(2 + \pi)}{2}$

Calculating this gives:

$m \approx -2.57$

Interpretation: The slope means that for an increase of 1 metre in the radius of the semicircle, the height of the rectangle falls by approximately 2.57 metres.

The area A of the Norman window is given by the sum of the area of the rectangle and the area of the semicircle:

• The area of the rectangle is given by: $A_{rectangle} = x \cdot y$
• The area of the semicircle is given by: $A_{semicircle} = \frac{1}{2}\pi r^2 = \frac{1}{2} \pi x^2$

Substituting y from above: $A = xy + \frac{1}{2} \pi x^2 = x \left(\frac{12 - (2 + \pi)x}{2}\right) + \frac{1}{2}\pi x^2$

By simplifying, we eventually arrive at: $a(x) = \frac{1}{2}(24x - (\pi + 4)x^2).$

To find the derivative of the area function: $a(x) = \frac{1}{2}(24x - (\pi + 4)x^2)$, we differentiate: $a'(x) = \frac{1}{2}(24 - 2(\pi + 4)x) = 12 - (\pi + 4)x.$ This expression describes the rate of change of the area with respect to changes in the radius x.