HELP Extremely difficult calculus problem

[don1231915]

HELP!Extremely difficult calculus problem! Optimising to find the maximum volume.

d(x)

 

[SEngstrom]

If you assume the cuboid's lateral dimensions to be (x,y), the point of contact at (x,y,z) with the ellipsoid will give you its height. Then given the various constraints you have - find (x,y) that maximizes the volume... There will be several points at the ends of these intervals that could be the maximum, but you have to look for local internal maxima as well.

 

[seto6]

where is the question? i don't see it

 

[SEngstrom]

Looks like the OP edited it and deleted it by mistake...

 

[don1231915]

where is the question? i don't see it

I have a upside down looking curve structure (½ ellipse). It has the following specifications:
The building has a rectangular base 150m long and 72m wide. The max height of the structure should not exceed 75% of its width or be less than half the width. And the min. height of a room in the building is 2.5m

The height of the structure is 36m
so acc. to my graph: http://imageupload.org/pt-112919260786.html

The equation is y= sqrt (1296-x^2)
How to find the dimensions of a cuboid with max. volume which fits inside this curve?

PLS REPLY

 

[don1231915]

If you assume the cuboid's lateral dimensions to be (x,y), the point of contact at (x,y,z) with the ellipsoid will give you its height. Then given the various constraints you have - find (x,y) that maximizes the volume... There will be several points at the ends of these intervals that could be the maximum, but you have to look for local internal maxima as well.

that method is really complex, i mean is there any other way rather than using 3d graphs?
Like, differentiating or something
also, isn't the local maximum just (0,36)

Thank you so much for helping me!

 

[SEngstrom]

If you let the axis of the ellipse be (a,b,c) and the cuboid have the size (2x,2y,z) - the point of contact is defined by
$$(x/a)^2+(y/b)^2+(z/c)^2=1$$
Maximize x*y*z with this constraint.

(x,y)=(0,36) would have zero volume so, no, that is not a maximum, local or otherwise.

 

[don1231915]

If you let the axis of the ellipse be (a,b,c) and the cuboid have the size (2x,2y,z) - the point of contact is defined by
$$(x/a)^2+(y/b)^2+(z/c)^2=1$$
Maximize x*y*z with this constraint.

(x,y)=(0,36) would have zero volume so, no, that is not a maximum, local or otherwise.

Cool, so that would just be x+y+z= 18, i can't download autograph (the only best 3d graphing software). I am super curious about how the graph of this might look like. Any ideas about the software?
That is a really in depth explanation and sounds difficult too. I actually wanted to keep it simple. So what I did was found out the volume
V= 300xy
i plugged in y with the equation of the ellipse (y = sqrt(1296-x^2)) and then differentiated and found x and therefore y and also the max volume.
Thank you so much for your help!
I have one final question, I know the max volume of the cuboid. I just need to know the volume of the structure (i.e. curve).
I know how to find the area using the integral but would that be the same as volume??

Thanks again