Respuesta :
Answer:
The dimensions of the box are 12 cm , 12 cm , 12 cm
Step-by-step explanation:
Let x , y and z be the dimensions of box
Volume of box =xyz=1728
[tex]z=\frac{1728}{xy}[/tex]
Surface area of box = [tex]2xy+2yz+2xz=2xy+2y(\frac{1728}{xy})+2x(\frac{1728}{xy})[/tex]
Let [tex]f(x,y)=2xy+2(\frac{1728}{x})+2(\frac{1728}{y})[/tex]
To get minimal surface area
[tex]\frac{\partial f}{\partial x}=0[/tex] and [tex]\frac{\partial f}{\partial y}=0[/tex]
[tex]\frac{\partial(2xy+2(\frac{1728}{x})+2(\frac{1728}{y}))}{\partial x}=0[/tex]
[tex]2y-2(\frac{1728}{x^2})=0[/tex]
[tex]y=\frac{1728}{x^2}[/tex] ----1
[tex]\frac{\partial(2xy+2(\frac{1728}{x})+2(\frac{1728}{y}))}{\partial y}=0[/tex]
[tex]2x-2(\frac{1728}{y^2})=0\\x=\frac{1728}{y^2} \\y^2=\frac{1728}{x}[/tex]
Using 1
[tex](\frac{1728}{x^2} )^2=\frac{1728}{x}[/tex]
x=0 and [tex]x^3=1728[/tex]
Side can never be 0
So,[tex]x^3=1728[/tex]
x=12
[tex]y=\frac{1728}{x^2} \\y=\frac{1728}{12^2}[/tex]
y=12
[tex]z=\frac{1728}{xy}\\z=\frac{1728}{(12)(12)}[/tex]
z=12
The dimensions of the box are 12 cm , 12 cm , 12 cm
Dimensions of side with minimum surface area is 12 cm
Given that;
Volume of box = 1728 cubic centimeter
Find:
Dimensions of side with minimum surface area
Computation:
Dimensions of side with minimum surface area is with cube
So,
Assume; Side = a
Volume of box = side³
Side³ = 1728
Side = ∛1728
Side = 12 cm
Learn more:
https://brainly.com/question/4694222?referrer=searchResults
