# Thread: Box optimization L, W, H, and domain

1. ## Box optimization L, W, H, and domain

A box is to be made out of a 12 cm by 18 cm piece of cardboard. Squares of side length cm will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. (a) Express the volume of the box as a function of .

(b) Give the domain of in interval notation. (Use the fact that length and volume must be positive.)

(c) Find the length , width , and height of the resulting box that maximizes the volume. (Assume that ).

= cm

= cm

= cm
(d) The maximum volume of the box is .

i found the equation and set the first derivative equal to zero and solved for x then plugging that number x into the original equation i found to get the max volume.

any help with finding the L, W, and H and the domain would be great.

thank you

2. So you have $\displaystyle L=(18-2x)$ and $\displaystyle W=(12-2x)$, and $\displaystyle H=x$ , since you found the x, just plug it in them.

For the domain you have 0 < x < 6 ...those are greater or equal by the way.

3. the L, W, and H were correct as you told me how to find them but the domain was (0,6) so i believe you were right before you said that they were equal to since 0 and 6 are not included =)

thank you very much.

4. Well, it doesn't really matter for this particular problem. But, since you are finding an absolute max, therefore by the Extreme Value Theorem the Volume attains its absolute max. at either the critical numbers or the end points: x=0 and x=6.
Therefore, I strongly believe it should be [0,6] instead of (0,6)

5. Originally Posted by Arturo_026
Well, it doesn't really matter for this particular problem. But, since you are finding an absolute max, therefore by the Extreme Value Theorem the Volume attains its absolute max. at either the critical numbers or the end points: x=0 and x=6.
Therefore, I strongly believe it should be [0,6] instead of (0,6)
thats what i think too because i entered [0,6] first because i believe the endpoints should be included but my webwork said that was incorrect and so i changed it to not include the endpoints (0,6) and it accepted that.

but i agree with you and i have found errors in my webwork assignments so this could just be one haha