Respuesta :
The cost of constructing the box will be = $81.77
Since the question gives us information about the rectangle,
so the area of the rectangle is shown by,
Area = 2 x (length + width)
So we have the area as,
Area = 2(lh + wh) +lw
(since the area per square meter is also given so we added l x w)
Since the cost of the base is = $5 per square meter and the cost of the sides is = $3 per square meter, so area in terms of the cost function,
⇒ C = 3 x 2(lh + wh) + 5 x l x w
⇒ C = 6(lh + wh) +5lw ---------equation 1
According to the question, the length of the rectangular container is twice that of the width of the container,
⇒ l = 2w
Substituting the new value of l in equation 1,
⇒ C = 6(2wh + wh) + 10 [tex]w^{2}[/tex]
⇒ C = 18wh + 10 [tex]w^{2}[/tex] ---------- equation 2
To calculate the volume of the rectangle,
V = length x width x height ( lwh )
substituting v = 10 and 2w = l,
2[tex]w^{2}[/tex]h = 10
h = [tex]\frac{5}{w^{2} }[/tex]
Substitute the value of h in equation 2,
⇒ C = 18w x [tex]\frac{5}{w^{2} }[/tex] + 10[tex]w^{2}[/tex]
⇒ C = [tex]\frac{90}{w}[/tex] + 10[tex]w^{2}[/tex] --------- equation 3
Differentiating the above equation we get,
⇒ C* = - [tex]\frac{90}{w^{2} }[/tex] + 20w
⇒ 20w = [tex]\frac{90}{w^{2} }[/tex]
Multiply both sides by [tex]w^{3}[/tex]
⇒ [tex]20w^{3}[/tex] = 90
⇒ [tex]w^{3}[/tex] = 4.5
w = 1.65
Put the value of w in equation 3,
⇒ C = [tex]\frac{90}{1.65}[/tex] + 10 x [tex]1.65^{2}[/tex]
⇒ C = 81.77
Therefore, The cost of constructing the box will be = $81.77
To know more about volumes,
https://brainly.com/question/22529898
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