MHB Optimizing the Area of a Norman Window - 25 Feet Perimeter

[MarkFL]

Here is the question:

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal?

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 25 feet?

Here is a link to the problem:

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal? - Yahoo! Answers

I have posted a link there to this topic so that the OP can find my response.

 

[MarkFL]

Hello Christopher,

One approach is to use Lagrange multipliers. We have the objective function:

$\displaystyle f(h,r)=2rh+\frac{1}{2}\pi r^2$

subject to the constraint:

$\displaystyle g(h,r)=2r+2h+\pi r-P=0$

giving the system:

$\displaystyle r=\lambda$

$\displaystyle 2h+\pi r=\lambda (2+\pi)$

which implies:

$\displaystyle h=r$

Substituting into the constraint, we then find:

$\displaystyle g(r)=(4+\pi)r=P$

$\displaystyle h=r=\frac{P}{4+\pi}$

And finally, substituting into the objective function, we find:

$\displaystyle f_{\text{max}}=\frac{P^2}{2(4+\pi)}$

Now, plug in the given perimeter for P to find the maximum area of the window:

$\displaystyle f_{\text{max}}=\frac{(25\text{ ft})^2}{2(4+\pi)}=\frac{625}{2(4+\pi)}\,\text{ft}^2$