Answer:
x=15.08 cm
y=7.54 cm
Step-by-step explanation:
We are given that
Volume of box=1715 cubic cm
Side length of base=x
l=b=x
Height of box=h=y
Volume of box=lbh=x^2y
[tex]1715=x^2y[/tex]
[tex]y=\frac{1715}{x^2}[/tex]
Surface area of box=Area of bottom+area of four faces=[tex]x^2+4xy[/tex]
[tex]S=x^2+4x(\frac{1715}{x^2}=x^2+\frac{6860}{x}[/tex]
Differentiate w.r.t x
[tex]S'(x)=2x-\frac{6860}{x^2}[/tex]
[tex]S'(x)=0[/tex]
[tex]2x-\frac{6860}{x^2}=0[/tex]
[tex]2x=\frac{6860}{x^2}[/tex]
[tex]x^3=\frac{6860}{2}=3430[/tex]
[tex]x=(3430)^{\frac{1}{3})=15.08[/tex]
Again differentiate w.r.t x
[tex]S''(x)=2+\frac{13720}{x^3}[/tex]
Substitute x=15.08
[tex]S''(x)=2+\frac{13720}{(15.08)^3}>0[/tex]
Hence, the surface area is minimum at x=15.08 cm
[tex]y=\frac{1715}{(15.08)^2}=7.54 cm[/tex]
