Respuesta :
Answer:
Dimensions will be
Length = 7.23 cm
Width = 7.23 cm
Height = 9.64 cm
Step-by-step explanation:
A closed box has length = l cm
width of the box = w cm
height of the box = h cm
Volume of the rectangular box = lwh
504 = lwh
[tex]h=\frac{504}{lw}[/tex]
Sides which involve length and width and height, cost = 3 cents per cm²
Top and bottom of the box costs = 4 cents per cm²
Cost of the sides [tex]C_{s}[/tex]= 3[2(l + w)h] = 6(l + w)h
[tex]C_{s}[/tex]= 3[2(l + w)h]
[tex]C_{s}=6(l+w)(\frac{504}{lw} )[/tex]
Cost of the top and the bottom [tex]C_{(t,p)}[/tex]= 4(2lw) = 8lw
Total cost of the box C = [tex]3024\frac{(l+w)}{lw}[/tex] + 8lw
= [tex]3024[\frac{1}{l}+\frac{1}{w}][/tex] + 8lw
To minimize the cost of the sides
[tex]\frac{dC}{dl}=3024(-l^{-2}+0)+8w=0[/tex]
[tex]\frac{3024}{l^{2}}=8w[/tex]
[tex]\frac{378}{l^{2}}=w[/tex] ---------(1)
[tex]\frac{dC}{dw}=3024(-w^{-2})+8l=0[/tex]
[tex]\frac{3024}{w^{2}}=8l[/tex]
[tex]\frac{378}{w^{2}}=l[/tex]
[tex]w^{2}=\frac{378}{l}[/tex]-------(2)
Now place the value of w from equation (1) to equation (2)
[tex](\frac{378}{l^{2}})^{2}=\frac{378}{l}[/tex]
[tex]\frac{(378)^{2} }{l^{4}}=\frac{378}{l}[/tex]
l³ = 378
l = ∛378 = 7.23 cm
From equation (2)
[tex]w^{2}=\frac{378}{7.23}[/tex]
[tex]w^{2}=52.28[/tex]
w = 7.23 cm
As lwh = 504 cm³
(7.23)²h = 504
[tex]h=\frac{504}{(7.23)^{2}}[/tex]
h = 9.64 cm
