Thread: Cuboid question.

A cuboid has a total surface area of 150cm^2 with a square base x cm.
Show that the height, h cm, of the cuboid is given by h=(75-x^2)/(2x).
Express the volume of the cuboid in terms of x.
Hence determine its maximum volume as x varies.

Originally Posted by grooverandshaker

A cuboid has a total surface area of 150cm^2 with a square base x cm.
Show that the height, h cm, of the cuboid is given by h=(75-x^2)/(2x).
Express the volume of the cuboid in terms of x.
Hence determine its maximum volume as x varies.

What have you tried?

To be completely honest, I have no idea where to start.
But for the second bit, I think I would do 150cm^2 * (75-x^2)/(2x) to find the volume, maybe??
And then for the third bit, I'll find the derivative, let it equal 0 to find x and once I find x sub it back into the volume equation.
I just don't really understand how to do the first bit.

Originally Posted by grooverandshaker

To be completely honest, I have no idea where to start.
But for the second bit, I think I would do 150cm^2 * (75-x^2)/(2x) to find the volume, maybe??
And then for the third bit, I'll find the derivative, let it equal 0 to find x and once I find x sub it back into the volume equation.
I just don't really understand how to do the first bit.

You have to do question 1 before question 2. How would you evaluate the total surface area of a cuboid?

Um, I'm not too sure.
Would it be 2x^2+4xh ...

Originally Posted by grooverandshaker

Um, I'm not too sure.
Would it be 2x^2+4xh ...

Yes. You are told that the total surface area is 150cm^2.

So that means 2x^2 + 4xh = 150. Solve this for h.

Ok so that'll end up being h=(75 - x^2)/2x.
Do I need to now sub this into something before I can start with the second question?

Right, now how would you evaluate the volume of the cuboid?

I don't really know...
The volume of the cuboid would maybe be h*l*w.
So I spose that would be x^2*h.
But, I don't think that's right.

Originally Posted by grooverandshaker

I don't really know...
The volume of the cuboid would maybe be h*l*w.
So I spose that would be x^2*h.
But, I don't think that's right.

Yes that's right. You have also found that h = (75 - x^2)/(2x). So substitute this in and you have the volume in terms of x.