Answer
A) $N(w) = 80w+\frac{\pi}{8}w^2$
B) $4395.22\ in^2$
Work Step by Step
Part A
Let $N$ be the total area of the Norman window. The area of the window is the sum of the rectangular part and the semi circle. The area of the half circle is just half the area of a complete circle.
$$N(w) = 80w+\frac{\pi}{8}w^2.\tag{1}$$ Part B
Set $w= 45$ in equation $(1)$ to find the area.
$$\begin{aligned}
N(45) &= 80\cdot 45+\frac{\pi}{8}\cdot 45^2\\
&\approx 4395.22\ in^2.
\end{aligned}$$
