Respuesta :
Answer:
- Base Length = 7.368cm
- Height = 0.737cm.
Step-by-step explanation:
Volume of the jewelry box=[tex]40cm^3[/tex]
The box has a square base and is to be built with silver plated sides and nickel plated top and base.
Therefore: Volume = Square Base Area X Height = l²h
[tex]l^2h=40\\h=\frac{40}{l^2}[/tex]
Total Surface Area of a Cuboid =2(lb+lh+bh)
Since we have a square base
Total Surface Area =[tex]2(l\²+lh+lh)[/tex]
The Total Surface Area of the box [tex]=2l\²+4lh[/tex]
Nickel plating costs $1 per [tex]cm\³[/tex]
Silver Plating costs $10 per [tex]cm\³[/tex]
Since the sides are to be silver plated and the top and bottom nickel plated:
Therefore, Cost of the Material for the jewelry box [tex]=1(2l\²)+10(4lh)[/tex]
[tex]Cost, C(l,h)=$(2l\²+40lh)[/tex]
Recall earlier that we derived: [tex]h=\frac{40}{l^2}[/tex]
Substituting into the formula for the Total Cost
[tex]Cost, C(l)=2l\²+40l(\frac{40}{l^2})\\=2l\²+\frac{1600}{l}\\C=\frac{2l^3+1600}{l}[/tex]
The minimum costs for the material occurs at the point where the derivative equals zero.
[tex]C^{'}=\frac{4l^3-1600}{l^2}[/tex]
[tex]4l^3-1600=0\\4l^3=1600\\l^3=400\\l=\sqrt[3]{400}=7.368 cm[/tex]
Recall:
[tex]h=\frac{40}{l^2}=\frac{40}{7.368^2}=0.737cm[/tex]
The box which minimizes the cost of materials has a square base of side length 7.368cm and a height of 0.737cm.
