Respuesta :
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
[tex]\Rightarrow z=\frac{19652}{xy}[/tex]......(2)
putting the value of z in the equation (1)
[tex]f(x,y)=xy+2x(\frac{19652}{xy})+2y(\frac{19652}{xy})[/tex]
[tex]\Rightarrow f(x,y)=xy+\frac{39304}{y})+\frac{39304}{x}[/tex]
The partial derivatives are
[tex]f_x=y-\frac{39304}{x^2}[/tex]
[tex]f_y=x-\frac{39304}{y^2}[/tex]
To find the dimension of the box set the partial derivatives [tex]f_x=0[/tex] and [tex]f_y=0[/tex].Therefore [tex]y-\frac{39304}{x^2}=0[/tex]
[tex]\Rightarrow y=\frac{39304}{x^2}[/tex].......(3)
and [tex]x-\frac{39304}{y^2}=0[/tex]
[tex]\Rightarrow x=\frac{39304}{y^2}[/tex].......(4)
Now putting the x in equation (3)
[tex]y =\frac {39304}{(\frac{39304}{y^2})^2}[/tex]
[tex]\Rightarrow y=\frac{y^4}{39304}[/tex]
[tex]\Rightarrow y^3= 39304[/tex]
⇒y=34 cm
Then [tex]\Rightarrow x=\frac{39304}{34^2}[/tex] =34 cm.
Putting the value of x and y in the equation (2)
[tex]z=\frac{19652}{34 \times 24}[/tex]
=17 cm.
The dimension of the cardboard is 34 cm by 34 cm by 17 cm.
