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.