Question

A designer is making a rectangular prism box with maximum volume, with the sum of its length, width, and height 8 in. The length must be 2
times the width. What should each dimension be? Round to the nearest tenth of an inch if necessary.

Answer 1

Answer:

width = 1.8 in

length =   3.6 in

height =$8-3w =8-3(1.8)= 2.6 in$

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

$l +w+h=8\\l=2w$

Plug it in the first equation

$2w+w+h=8\\3w+h=8\\h=8-3w$

volume the box is length times width times height

$volume = (2w)(w)(8-3w)\\16w^2-6w^3$

To get maximum volume we take derivatived

$v'=32w-18w^2$

set the derivative =0 and solve for w

$0=32w-18w^2\\0=-2w(9w-16)\\w=0 , w= \frac{16}{9}$

width = 1.8 in

length = 2w= 3.6 in

height =$8-3w =8-3(1.8)= 2.6 in$