The bag's greatest volume is 5184 metric 3
The given parameters are:
l+ w+ h = 54
l is length, b is breadth, h is height
l = 2h
Substitute l = 2h in l+w+h = 54
2h+w+h=54
This gives
3h+w = 54
Now w is
w=54-3h
The volume of the bag is
V= lwh
Substitute l=2h
V= 2wh²
Substitute w=54-3h
V =2(54 — 3h)h²
V = 108h² —6h³
Differentiate
V' = 216h — 18h²
Equate to 0
216h — 18h² =0
Factorize
h(216-18h) = 0
h = 0, 12
The height cannot be 0.
So, we have:
h=12
Substitute h in above equations we get l and w as
w=54-3h = 54-12(3) = 54-36= 18
l = 54-h-w = 54-12-18= 24
We can find volume by using,
V = lwh;
V = 18 * 24 * 12 = 5184
Hence, the largest volume of the bag is 5184 metric ³
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