# Lagrange multipliers help

1. Dec 23, 2008

### david12

help me out on this proble i am confuse

a sport center is to be constructed.it consists of a rectangular region with a semicircle ach end .if the perimater of the room is to be a 500 meter running truck find the dimetion that will make the area as large as possible.

i can find if the picture only has a rectange.but the perimater of the half circle which is pir^2

p = 2lw + PI r

500=2lw + pi r

2. Dec 23, 2008

### Defennder

Re: calc

Hmm you can use Lagrange multipliers to solve this but it may be unnecessary. Otherwise you can just find the values of l,r (note that w=2r) such that Area is maximised. You wrote p when it should meant Area instead. So find the stationary points of the function A(r,l) and check which is a max point.

3. Dec 23, 2008

### HallsofIvy

Staff Emeritus
Re: calc

No. First, the perimeter of a rectangle is 2l+ 2w, not "2lw" (you are mixing perimeter and area formulas). Also if you run around the half circle you do NOT run along that end of the rectangle. Taking the semi-circle to be on the "w" end, the distance is 2l+ w+ pi r.

500= 2l+ w+ pi rl. Also, have you drawn a picture of this? Is so you should have been able to see that the semi-circle has one end (w) of the rectangle as diameter: r= 2w.

You want to maximize the area. What is the formula for the area, as a function of l and w? You can use the perimeter formula to eliminate one of the variables.

Edit: Does "with a semicircle ach end" mean a semi-circle at each end? That makes more sense! I was wondering about runing around the corners! In that case, there is no "w" length so the perimeter is 2l+ 2pi r. And, of course, since we now have an entire circle added the area is lw+ pi r^2= 2lr+ pi r^2.

Last edited: Dec 24, 2008
4. Dec 23, 2008

### Dick

Re: calc

What does 'lw' mean? The perimeter of a semicircle is pi*r (not pi*r^2), but there are two of them in the perimeter. You'll really need to be a lot clearer about what you are doing. Once you said what all the symbols mean, what's a formula for the area?