Answer:
[tex]x = y = z = 20[/tex]
Step-by-step explanation:
Given
[tex]Volume = 8000cm^3[/tex]
Required
Determine the dimensions that minimizes the surface area.
Surface area of a box is;
[tex]S = 2(xy + xz + yz)[/tex]
Volume of a box is:
[tex]V = xyz[/tex]
Make z the subject
[tex]z = \frac{V}{xy}[/tex]
Substitute 8000 for V
[tex]z = \frac{8000}{xy}[/tex]
Substitute 1000/xy for z in [tex]S = 2(xy + xz + yz)[/tex]
[tex]S = 2(xy + x\frac{8000}{xy}+y\frac{8000}{xy})[/tex]
[tex]S = 2(xy + \frac{8000}{y}+\frac{8000}{x})[/tex]
Expand:
[tex]S = 2xy + \frac{16000}{y}+\frac{16000}{x}[/tex]
[tex]S = 2xy + 16000(\frac{1}{y}+\frac{1}{x})[/tex]
Differentiate S w.r.t x
[tex]\frac{dS}{dx} = 2y - \frac{16000}{x^2}[/tex]
Differentiate S w.r.t y
[tex]\frac{dS}{dy} = 2x - \frac{16000}{y^2}[/tex]
Equate both differentiation to 0
[tex]2x - \frac{16000}{y^2} = 0[/tex]
Multiply through by [tex]y^2[/tex]
[tex]2xy^2 - 16000 = 0[/tex]
[tex]2xy^2 = 16000[/tex]
Divide through by 2
[tex]xy^2 = 8000[/tex] -- (1)
[tex]2y - \frac{16000}{x^2} = 0[/tex]
Multiply through by x^2
[tex]2x^2y - 16000 = 0[/tex]
[tex]2x^2y = 16000[/tex]
Divide through by 2
[tex]x^2y = 8000[/tex] --- (2)
Divide (1) by (2)
[tex]\frac{x^2y = 8000}{xy^2 = 8000}[/tex]
[tex]\frac{x}{y} = 1[/tex]
[tex]x = y[/tex]
Substitute y for x in (1)
[tex]x^2y = 8000[/tex]
[tex]x^2 * x = 8000[/tex]
[tex]x^3 = 8000[/tex]
Take cube roots
[tex]x =20[/tex]
Hence;
[tex]x = y = 20[/tex]
Recall that:
[tex]z = \frac{8000}{xy}[/tex]
[tex]z = \frac{8000}{20 * 20}[/tex]
[tex]z = 20[/tex]
Hence, the dimension of the box that minimizes the surface area of the box is:
[tex]x = y = z = 20[/tex]
