A box with a square base and no top is to be built with a volume of 1638416384 in33. Find the dimensions of the box that requires the least amount of material. How much material…

Question

asked May 6, 2022 180k views

1 Answer

Atzuki · Answer 1 · 2022-05-11T15:07:29+0000

Answer:

$512\ \text{in}^2$

Explanation:

x = Length and width of base

y = Height of box

Volume of the box is
$16384\ \text{in}^3$

$x^2y=16384\\\Rightarrow y=(16384)/(x^2)$

Surface area is given by

$s=x^2+4y\\\Rightarrow s=x^2+4* (16384)/(x^2)\\\Rightarrow s=x^2+(65536)/(x^2)$

Differentiating with respect to x we get

s'=2x-(131072)/(x^3)

Equating with 0 we get

$0=2x^4-131072\\\Rightarrow x=((131072)/(2))^{(1)/(4)}\\\Rightarrow x=16$

s''=2+(393216)/(x^4)

at
x=16

s''=2+(393216)/(16^4)=8>0

So the function is minimum at x = 16

$y=(16384)/(x^2)=(16384)/(16^2)\\\Rightarrow y=64$

The material required is

$s=x^2+4y=16^2+4* 64\\\Rightarrow s=512\ \text{in}^2$

The minimum amount of material required is
$512\ \text{in}^2$ .