Answer :
Answer:
width = 1.8 in
length = 3.6 in
height =[tex]8-3w =8-3(1.8)= 2.6 in[/tex]
Step-by-step explanation:
A designer is making a rectangular prism box with maximum volume, with the sum of its length, width, and height 8 in
Let l , w and h are the length , width and height
[tex]l +w+h=8\\l=2w[/tex]
Plug it in the first equation
[tex]2w+w+h=8\\3w+h=8\\h=8-3w[/tex]
volume the box is length times width times height
[tex]volume = (2w)(w)(8-3w)\\16w^2-6w^3[/tex]
To get maximum volume we take derivatived
[tex]v'=32w-18w^2[/tex]
set the derivative =0 and solve for w
[tex]0=32w-18w^2\\0=-2w(9w-16)\\w=0 , w= \frac{16}{9}[/tex]
width = 1.8 in
length = 2w= 3.6 in
height =[tex]8-3w =8-3(1.8)= 2.6 in[/tex]