Answer:
- Base Length = 7.368cm
- Height = 0.737cm.
Step-by-step explanation:
Volume of the jewelry box=![40cm^3](https://tex.z-dn.net/?f=40cm%5E3)
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
![l^2h=40\\h=\frac{40}{l^2}](https://tex.z-dn.net/?f=l%5E2h%3D40%5C%5Ch%3D%5Cfrac%7B40%7D%7Bl%5E2%7D)
Total Surface Area of a Cuboid =2(lb+lh+bh)
Since we have a square base
Total Surface Area =![2(l\²+lh+lh)](https://tex.z-dn.net/?f=2%28l%5C%C2%B2%2Blh%2Blh%29)
The Total Surface Area of the box ![=2l\²+4lh](https://tex.z-dn.net/?f=%3D2l%5C%C2%B2%2B4lh)
Nickel plating costs $1 per ![cm\³](https://tex.z-dn.net/?f=cm%5C%C2%B3)
Silver Plating costs $10 per ![cm\³](https://tex.z-dn.net/?f=cm%5C%C2%B3)
Since the sides are to be silver plated and the top and bottom nickel plated:
Therefore, Cost of the Material for the jewelry box ![=1(2l\²)+10(4lh)](https://tex.z-dn.net/?f=%3D1%282l%5C%C2%B2%29%2B10%284lh%29)
![Cost, C(l,h)=$(2l\²+40lh)](https://tex.z-dn.net/?f=Cost%2C%20C%28l%2Ch%29%3D%24%282l%5C%C2%B2%2B40lh%29)
Recall earlier that we derived: ![h=\frac{40}{l^2}](https://tex.z-dn.net/?f=h%3D%5Cfrac%7B40%7D%7Bl%5E2%7D)
Substituting into the formula for the Total Cost
![Cost, C(l)=2l\²+40l(\frac{40}{l^2})\\=2l\²+\frac{1600}{l}\\C=\frac{2l^3+1600}{l}](https://tex.z-dn.net/?f=Cost%2C%20C%28l%29%3D2l%5C%C2%B2%2B40l%28%5Cfrac%7B40%7D%7Bl%5E2%7D%29%5C%5C%3D2l%5C%C2%B2%2B%5Cfrac%7B1600%7D%7Bl%7D%5C%5CC%3D%5Cfrac%7B2l%5E3%2B1600%7D%7Bl%7D)
The minimum costs for the material occurs at the point where the derivative equals zero.
![C^{'}=\frac{4l^3-1600}{l^2}](https://tex.z-dn.net/?f=C%5E%7B%27%7D%3D%5Cfrac%7B4l%5E3-1600%7D%7Bl%5E2%7D)
![4l^3-1600=0\\4l^3=1600\\l^3=400\\l=\sqrt[3]{400}=7.368 cm](https://tex.z-dn.net/?f=4l%5E3-1600%3D0%5C%5C4l%5E3%3D1600%5C%5Cl%5E3%3D400%5C%5Cl%3D%5Csqrt%5B3%5D%7B400%7D%3D7.368%20cm)
Recall:
![h=\frac{40}{l^2}=\frac{40}{7.368^2}=0.737cm](https://tex.z-dn.net/?f=h%3D%5Cfrac%7B40%7D%7Bl%5E2%7D%3D%5Cfrac%7B40%7D%7B7.368%5E2%7D%3D0.737cm)
The box which minimizes the cost of materials has a square base of side length 7.368cm and a height of 0.737cm.