Maximizing Volume: Rectangular Box in Hemisphere | Homework Problem

[bodensee9]

Homework Statement

Hello:

This is a max/min problem. I am asked to find the rectangular box of maximum volume inscribed in a hemisphere of radius R.

Homework Equations

So I am wondering if I have set up this correctly. If say my length is x, width is y, and height is z. So, I would have max(xyz). And then would I have R^{2}-\frac{x^{2}}{4}-\frac{y^{2}}{4} = z^{2} I am not sure if this is the correct relationship. And then my function would be: f(x, y) = xy(r^{2}-\frac{x^{2}}{4}-\frac{y^{2}}{4})

I think I can maximize xyz using xyz^2.

thanks!

 

[scottie_000]

Firstly, what method are you aiming to use in order to maximize xyz? You need to be clear in your own mind what you are going to do with f.

Setting up the problem is usually a matter of defining your constraints. i.e.
What hemisphere is your constraint R^{2}-\frac{x^{2}}{4}-\frac{y^{2}}{4} = z^{2} describing?
Are x,y,z all positive or can some be negative?

Also, before you get going in these problems, do you have a feel for what the correct answer might be?

 

[bodensee9]

it's a hemisphere, I would think that z would be all positive because I am assuming that it's the upper hemisphere. I am going to use either Lagrange multiplier's method or the usual gradient = 0 method. I am sure the answer will be some multiple of R.

 

[scottie_000]

Yes the answer is likely to involve R somewhere. However, your equation for the hemisphere is slightly off: can you write down the equation for a sphere of radius R in terms of x,y and z?
Once you have that, it is a simple case of restricting z to be positive as you said.

Also, maximizing the volume, V, is equivalent to maximizing V^2 so to simplify algebra you can take x^2 y^2 z^2 rather than xyz^2 which is not necessarily the same

 

[bodensee9]

Hello:

Thanks, so would the equation for a sphere be:
x^{2} + y^{2} + z^{2} = R^{2}
So then my constraint is z^{2} = R^{2}-x^{2}-y^{2}?
And then my function would be to maximize:
x^{2}y^{2}(R^{2}-x^{2}-y^{2})?
Thanks.