window area

[john3j]

Can anybody guide me here or list the steps I need to take to be able to solve this? I would like to do the work on my own, so any guidance would be greatly appreciated!

A front window on a new home is designed as a rectangle with a semicircle on the top. It the window is designed to let in a maximum amount of light, and the architect fixes the perimeter of the entire window at 600 inches, determine the radius r and rectangular height so as to maximize the area.

Would I work this one as suggested at http://answers.yahoo.com/question/index?qid=20091114110720AAPQsVG ? I understand these instructions, but just want to make sure that I follow the correct procedure.

Thanks,
John

 

[MarkFL]

Another approach is Lagrange multipliers.

We have the objective function:

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

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:

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

\(\displaystyle h=r=\dfrac{P}{4+\pi}\)

Now, plug in the given perimeter for P to find the dimensions that maximize the area of the window.

 

[JeffM]

John

One of the most important things to do is to explain something about your background in math. It is very hard to answer questions intelligently when you have very little clue as to your audience.

If you are just starting calculus, you are unlikely to have learned yet about LaGrangian multipliers. If you are completing your first semester, LaGrangian multipliers may have been explained; if not, you should be able to grasp them quickly. How do we know your situation unless you tell us.