Question

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 is required at the minimum

Answer 1

Answer:

$512\ \text{in}^2$

Step-by-step explanation:

x = Length and width of base

y = Height of box

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

$x^2y=16384\\\Rightarrow y=\dfrac{16384}{x^2}$

Surface area is given by

$s=x^2+4y\\\Rightarrow s=x^2+4\times \dfrac{16384}{x^2}\\\Rightarrow s=x^2+\dfrac{65536}{x^2}$

Differentiating with respect to x we get

$s'=2x-\dfrac{131072}{x^3}$

Equating with 0 we get

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

$s''=2+\dfrac{393216}{x^4}$

at $x=16$

$s''=2+\dfrac{393216}{16^4}=8>0$

So the function is minimum at x = 16

$y=\dfrac{16384}{x^2}=\dfrac{16384}{16^2}\\\Rightarrow y=64$

The material required is

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

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