Chapter 12.8, Problem 19HP

To determine

To find : Least amount of cardboard possible and the dimension of box.

Answer to Problem 19HP

  $7.54\text{ cm, 7}\text{.54 cm, 3}\text{.79 cm}$

Explanation of Solution

Given information :

The given information is, A box manufacturer wants to make a box of volume $216{\text{ cm}}^{\text{3}}$ and uses least amount of cardboard.

Formula Used:

  $A=xy+2yz+2zx$ and $V=xyz$ .

Calculation :

  $216=xyz\Rightarrow z=\frac{216}{xy}$......(1)

  

  $A=xy+2y\left(\frac{216}{xy}\right)+2z\left(\frac{216}{xy}\right),$

Using (1)

  $A=xy+\frac{432}{x}+\frac{432}{y}$.....(2)

For least amount of cardboard, differentiating (2)

  $\begin{array}{l}{\frac{dA}{dx}|}_{y(\text{constant})}=y-\frac{432}{x^2}\\ \frac{dA}{dx}=0\Rightarrow y-\frac{432}{x^2}=0\left[\because \frac{dA}{dx}=0,\text{ for min}\text{. or max}\text{.}\right]\end{array}$

  $\Rightarrow y=\frac{432}{x^2}=432x^{-2}$.....(3)

Using (2),

  $\frac{dA}{dy}=0\Rightarrow x-\frac{432}{y^2}=0$

  $\Rightarrow x=\frac{432}{y^2}=432y^{-2}$.....(4)

From (3) and (4),

  $y=432{(432y^{-2})}^{-2}$ , Substituting from (4)

  $\begin{array}{l}\Rightarrow y=432\cdot {432}^{-2}{y}^4\\ \Rightarrow 432=y^3\\ $ \Rightarrow y={(432)}^{1/3}=7.54\text{ cm}$\end{array}$

Similarly, $x=7.54\text{ cm}$

  $z=\frac{216}{(7.54)\times (7.54)}=3.79\text{ cm}$

Dimensions of cardboard, $x=7.54\text{ cm,}y=7.54\text{ cm}$ and $z=3.79\text{ cm}$

Note : $\frac{d^{2}A}{d{x}^2}$ is positive and $\frac{d^{2}A}{d{y}^2}$ is positive.

So, these value are points of minima.