Question

A cardboard box without a lid is to have a volume of 19,652 cm3. Find the dimensions that minimize the amount of cardboard used. (Let x, y, and z be the dimensions of the cardboard box.)

Answer 1

Answer:

The dimension of the cardboard is 34 cm by 34 cm by 17 cm.

Step-by-step explanation:

Let the dimension of the cardboard box be x cm by y cm by z cm.

The surface area of the cardboard box without lid is

f(x,y,z)= xy+2xz+2yz.....(1)

Given that the volume of the cardboard is 19,652 cm³.

Therefore xyz =19,652

$\Rightarrow z=\frac{19652}{xy}$......(2)

putting the value of z in the equation (1)

$f(x,y)=xy+2x(\frac{19652}{xy})+2y(\frac{19652}{xy})$

$\Rightarrow f(x,y)=xy+\frac{39304}{y})+\frac{39304}{x}$

The partial derivatives are

$f_x=y-\frac{39304}{x^2}$

$f_y=x-\frac{39304}{y^2}$

To find the dimension of the box set the partial derivatives $f_x=0$ and $f_y=0$.Therefore $y-\frac{39304}{x^2}=0$

$\Rightarrow y=\frac{39304}{x^2}$.......(3)

and   $x-\frac{39304}{y^2}=0$

$\Rightarrow x=\frac{39304}{y^2}$.......(4)

Now putting the x in equation (3)

$y =\frac {39304}{(\frac{39304}{y^2})^2}$

$\Rightarrow y=\frac{y^4}{39304}$

$\Rightarrow y^3= 39304$

⇒y=34 cm

Then $\Rightarrow x=\frac{39304}{34^2}$ =34 cm.

Putting the value of x and y in the equation (2)

$z=\frac{19652}{34 \times 24}$

  =17 cm.

The dimension of the cardboard is 34 cm by 34 cm by 17 cm.