Norman Window area

[Dorian Gray]

Hello Mathematicians,

I have run into more issues with a different optimization problem.

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 feet, find the dimensions of the window so that the greatest possible amount of light is admitted.

I attached a copy of my work.

http://www.flickr.com/photos/77835031@N02/6858707594/

http://www.flickr.com/photos/77835031@N02/6858707940/


Please look at both if you have the chance. I did something slightly different between the two.


Thank you for any and all input.

 

[tkhunny]

Which one is right and complete?

 

[pappus]

Hello Mathematicians,

I have run into more issues with a different optimization problem.

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 feet,

...

The perimeter of this window is

\(\displaystyle p = 30 = 2L + W + \frac12 \pi W\)

 

[Dorian Gray]

thanks

Thank you both TKHunny and Pappas for your input. It is greatly appreciated.

@TKHunny. Neither of my pictures are correct. They are close, but not correct. The correct answers are \(\displaystyle 60/(4+pi) and 30/(4+pi)\)

I redid my work according to Pappus's perimeter equation, but I still did not come up with the correct answer.

Here is my work redone according to Pappus's perimeter equation.

 

[galactus]

Let r be the radius of the semicircle and let h be the height of the rectangle.

Then, the area of the window is \(\displaystyle A=2rh+\frac{{\pi}r^{2}}{2}\)

The perimeter is \(\displaystyle P=2r+2h+\pi r\)

\(\displaystyle h=\frac{1}{2}(p-(2+\pi)r)\)

So, \(\displaystyle A=pr-(2+\frac{\pi}{2})r^{2}\)

\(\displaystyle \frac{dA}{dr}=p-(4+\pi)r=0\)

when \(\displaystyle r=\frac{p}{4+\pi}\)

\(\displaystyle \frac{d^{2}A}{dr^{2}}<0\), so A is a max when:

\(\displaystyle r=\frac{p}{4+\pi}\)

 

[Dorian Gray]

thank you

Thank you Galactus!

I see how you addressed the formulas for the primary and secondary equations, which helped tremendously.