Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 56

The figure below is an example of Norman window which has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 feet, express the area $A$ of the window as a function the width $x$ of the window.








The perimeter of the window is equal to the sum of the perimeter of the rectangle and the circumference of the surmounted semicircle. Also, the radius of the circle could be represented as $\displaystyle \frac{x}{2}$. Which shows in the equation below:



$
\begin{equation}
\begin{aligned}

\displaystyle\text{Perimeter} &= x + 2y + \frac{\pi x}{2} = 30 && (\text{Solving for } y)\\
\\
y &= \frac{30 -x - \frac{\pi x}{2}}{2}
\end{aligned}
\end{equation}
$




On the other hand, the total area of the window is the sum of the area of the rectangle and the surmounted semicircle. Which is also shown on the equation below:



$\displaystyle \text{Area} = xy + \frac{\pi (\frac{x}{2})^2}{2}$


We can now substitute the value of $y$ in the equation of area to express the area in terms of the width $x$




$
\begin{equation}
\begin{aligned}

\text{Area} &= x \left( \frac{30-x-\frac{\pi}{2}x}{2}\right) + \frac{\pi x ^2}{8} && (\text{Simplify the equation})\\
\text{Area} &= 15x - \frac{x^2}{2} - \frac{x^2 \pi}{4} + \frac{\pi x ^2}{8} && (\text{Combine like terms})\\
\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

\fbox{$\text{ Area }= 15x - 0.90x^2$}
\end{aligned}
\end{equation}
$