The volume of the bag is the amount of space in it.
The largest volume of the bag is 9366
The given parameters are:
[tex]\mathbf{l +w+h =58}[/tex]
[tex]\mathbf{l = 2h}[/tex]
Substitute [tex]\mathbf{l = 2h}[/tex] in [tex]\mathbf{l +w+h =58}[/tex]
[tex]\mathbf{2h +w+h =58}[/tex]
This gives
[tex]\mathbf{3h +w =58}[/tex]
Make w the subject
[tex]\mathbf{w =58 - 3h}[/tex]
The volume of the bag is:
[tex]\mathbf{V = lwh}[/tex]
Substitute [tex]\mathbf{l = 2h}[/tex]:
[tex]\mathbf{V = 2wh^2}[/tex]
Substitute [tex]\mathbf{w =58 - 3h}[/tex]
[tex]\mathbf{V = 2(58 - 3h)h^2}[/tex]
[tex]\mathbf{V = (116 - 6h)h^2}[/tex]
Expand
[tex]\mathbf{V = 116h^2 - 6h^3}[/tex]
Differentiate
[tex]\mathbf{V' = 232h- 18h^2}[/tex]
Set to 0
[tex]\mathbf{ 232h- 18h^2 = 0}[/tex]
Factorize
[tex]\mathbf{ h(232- 18h) = 0}[/tex]
Split
[tex]\mathbf{ h = 0\ or\ 232- 18h = 0}[/tex]
The height cannot be 0.
So, we have:
[tex]\mathbf{ 232- 18h = 0}[/tex]
This gives
[tex]\mathbf{ 18h = 232}[/tex]
Divide both sides by 18
[tex]\mathbf{ h = 13}[/tex]
Recall that: [tex]\mathbf{w =58 - 3h}[/tex]
So, we have:
[tex]\mathbf{w =58 - 3(13)}[/tex]
[tex]\mathbf{w =19}[/tex]
Substitute 19 for w in [tex]\mathbf{l = 2h}[/tex]
[tex]\mathbf{l = 2(19)}[/tex]
[tex]\mathbf{l = 38}[/tex]
Recall that: [tex]\mathbf{V = lwh}[/tex]
So, we have:
[tex]\mathbf{V = 38 \times 19 \times 13}[/tex]
[tex]\mathbf{V = 9366}[/tex]
Hence, the largest volume of the bag is 9366
Read more about volumes at:
https://brainly.com/question/2198651
