Answer:
28cm*28cm*14cm
Step-by-step explanation:
Five side box for cardboard are: xy, 2xz and 2yz
so the function of Area will be:
[tex]f(x,y.z)=xy+2xz+2yz[/tex] equation 1
The Volume will be:
[tex]xyz=10976 cm^{3}[/tex] equation 2
we can make z the subject of the formula by divide both side by xy
now z = [tex]\frac{10976}{xy}[/tex] equation 3
To eliminate z, substitute z into equation 1
f(x,y)=xy+2x[tex](\frac{10976}{xy})[/tex]+2y
f(x,y)=xy+[tex]\frac{21952}{y}[/tex]+[tex]\frac{21952}{x}[/tex]
Now we have to derivative of x and y
[tex]f_{x}= y-\frac{21952}{x^{2} }[/tex]
[tex]f_{y}=x-\frac{21952}{y^{2} }[/tex]
for y and x
[tex]y-\frac{21952}{x^{2} } =0\\y=21952x^{-2} \\x-\frac{21952}{y^{2} } =0\\x=21952y^{-2}[/tex]
substitute y into x
[tex]x=21952(21952x^{-2} )^{-2} \\x=(21952)^{-1} x^{4} \\ x^{3} =21952\\cube-root-both side\\x=\sqrt[3]{21952} \\x=28[/tex]
to find y
[tex]y=21952(28)^{-2} \\y=28[/tex]
to find z
[tex]z=\frac{10976}{xy}[/tex]
z=[tex]\frac{10976}{28*28}[/tex]
[tex]z=14[/tex]
so The dimensions are 28cm*28cm*14cm
