Respuesta :
Answer:
6.3 cm by 6.3 cm by 12.6cm
Step-by-step explanation:
Volume of the box=[tex]500 cm^3[/tex]
The minimal dimensions of a box always occur when the base is a square.
[tex]L^2H=[/tex][tex]500 cm^3[/tex]
[tex]H=\frac{500}{L^2}[/tex]
Surface Area of a cylinder=[tex]2(L^2+LH+LH)[/tex]
Surface Area of the sides and bottom= [tex]L^2+2(LH+LH)[/tex]
Surface Area for the top = [tex]L^2[/tex]
The material for the sides and bottom costs $0.05 per [tex]cm^2[/tex]
The material for the top costs $0.15 per [tex]cm^2[/tex]
Therefore Cost of the box
[tex]C=0.15L^2+0.05[L^2+4LH]\\C=0.2L^2+0.2LH[/tex]
Recall:[tex]H=\frac{500}{L^2}[/tex]
[tex]C=0.2L^2+0.2L(\frac{500}{L^2})\\=0.2L^2+\frac{100}{L}\\C=\frac{0.2L^3+100}{L}[/tex]
The minimum value of C is at the point where the derivative is zero.
[tex]C^{'}=\frac{2(L^3-250)}{5L^2}\\\frac{2(L^3-250)}{5L^2}=0\\2(L^3-250)=0\\L^3=250\\L=6.3cm[/tex]
[tex]H=\frac{500}{L^2}=\frac{500}{6.3^2}=12.6cm[/tex]
The dimensions that would minimize the cost are 6.3 cm by 6.3 cm by 12.6cm
