Question

An open-top box with a square base is to be constructed from 120 square centimeters of material. What dimensions will produce a box with the largest possible volume

Answer 1

Answer:

Maximum length and breadth of the box is 6.32 cm and the height of the box is 3.16 cm

Step-by-step explanation:

Surface area of box

$4xy+x^2=120\\\Rightarrow y=\dfrac{120-x^2}{4x}$

Volume of box is

$V=x^2y\\\Rightarrow V=x^2\times \dfrac{120-x^2}{4x}\\\Rightarrow V=\dfrac{120x-x^3}{4}\\\Rightarrow V=30x-\dfrac{1}{4}x^3$

Differentiating with respect to $x$

$\dfrac{dV}{dx}=30-\dfrac{3}{4}x^2$

Equating with zero

$30-\dfrac{3}{4}x^2=0\\\Rightarrow x^2=\dfrac{30\times 4}{3}\\\Rightarrow x^2=40\\\Rightarrow x=6.32$

Double derivative of the volume

$\dfrac{d^2V}{dx^2}=-\dfrac{6x}{4}=-\dfrac{6\times 6.32}{4}=-9.48$

So, the volume is maximum at $x=6.32$

$y=\dfrac{120-x^2}{4x}=\dfrac{120-40}{4\times \sqrt{40}}\\\Rightarrow y=3.16$

So, the maximum length and breadth of the box is 6.32 cm and the height of the box is 3.16 cm.