Answer:
Side of square bottom=6.46 m
Height of box=4.31 m
Step-by-step explanation:
Let s be the side bottom
Height of box=h
Volume of box=[tex]180m^3[/tex]
Volume of box=[tex]lbh=s^2h[/tex]
[tex]s^2h=180[/tex]
[tex]h=\frac{180}{s^2}[/tex]
Cost of bottom=$40 per square m
Cost of sides =$30 per square m
Total cost=[tex]C=40s^2+30(4sh)[/tex]
[tex]C=40s^2+120s\times \frac{180}{s^2}[/tex]
[tex]C(s)=40s^2+\frac{21600}{s}[/tex]
Differentiate w.r.t s
[tex]C'(s)=80s-\frac{21600}{s^2}[/tex]
[tex]C'(s)=0[/tex]
[tex]80s-\frac{21600}{s^2}=0[/tex]
[tex]80s=\frac{21600}{s^2}[/tex]
[tex]s^3=\frac{21600}{80}=270[/tex]
[tex]s=(270)^{\frac{1}{3}}=6.46[/tex]
Again,differentiate w.r.t s
[tex]C''(s)=80+\frac{43200}{s^3}[/tex]
Substitute s=6.46
[tex]C''(6.46)=80+\frac{43200}{(6.46)^3}=240.2>0[/tex]
Hence, the cost is minimum at s=6.46
[tex]h=\frac{180}{s^2}=\frac{180}{(6.46)^2}=4.31[/tex]
Side of square bottom=6.46 m
Height of box=4.31 m
