For starters, I think you'll find that the length of the string is actually 6x + 2y...
I think that in the problem you solved, the rope length is 4x + 2y, not 4x + 4y. Correspondingly, L = 4x + 2V / x^2, not 4x + V / x^2. Otherwise, I agree.
In the first problem, the variables m, w, and h seem independent, so the rope length has to be expressed through all three. Since V = 2mwh, you can express one of m, w, h through V and the other two and so express L through V and two of m, w, h. In particular, I don't see how you can find L in terms of V and w only. Also, minimizing the rope length requires some other constraint (e.g., keeping the volume constant); otherwise, you can get L = 0 when m = w= h = 0 .
First, I get
L = 2(2h + 4m + 2w + 2h) = 8h + 8m + 4w. (*)
Here 2h corresponds to the front side, 4m + 2w to the top side, and 2h to the left side.
Second, V = 2mwh; substituting w = 3m, V = 6m^2h. Since also V = 3m^3, we get 2h = m. So, the side lengths are h, m = 2h and w = 3m = 6h. Also, V = 2mwh = 2 * 2h * 6h * h = 24h^3. So, everything can be expressed through h.
If you still need to find L in terms of w, then from (*) I get L = 8h + 8m + 4w = (8/6)w + (8/3)w + 4w.
Sorry to bother you further....
THe question asks to find the Length in terms of Width and Volume
i don't think this applies to you above solution or am i mistaken?
(oh and yes i do verify my all the calculations i have been nutting this question out on paper myself guided by your ideas in your threads)
Well, L = 8w satisfies this requirement; it just does not use volume.THe question asks to find the Length in terms of Width and Volume
You gave two additional equations: volume = 3m^3 and w = 3m, which allowed reducing the number of independent variables. Initially, there are 5 variables: m, w, h, V, L and 2 equations: V = 2mwh and L = 8h + 8m + 4w. The number of independent variables is the total number of variables minus the number of equations (that can't be derived from each other), so initially there are 3 independent variables. When you added V = 3m^3 and w = 3m, the number of independent variables was reduced to 1.
V = 2mwh = 2 * (w/3) * w * (w/6) = w^3 / 9. Here is the graph in WolframAlpha.