Answer:
The box which minimizes the cost of materials has a square base of side length 4.45cm and a height of 2.22 cm.
Step-by-step explanation:
Volume of the jewellery box=44cm³
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
l²h=44
h=44/l²
Total Surface Area of a Cuboid =2(lb+lh+bh)
Since we have a square base
Total Surface Area =2(l²+lh+lh)
The Total Surface Area of the box =2l²+4lh
Nickel plating costs $1 per cm³
Silver Plating costs $2 per cm³
Since the sides are to be silver plated and the top and bottom nickel plated:
Therefore, Cost of the Material for the jewellery box =1(2l²)+2(4lh)
Cost, C(l,h)=$(2l²+8lh)
Recall earlier that we derived:
h=44/l²
Substituting into the formula for the Total Cost
Cost, C(l)=2l²+8l(44/l²)
C=2l²+352/l
C=(2l³+352)/l
The minimum costs for the material occurs at the point where the derivative equals zero.
C'=(4l³-352)/l²
4l³-352=0
4l³=352
Divide both sides by 4
l³=88
l=4.45cm
Recall:
h=44/l²=44/4.45²=2.22cm
The box which minimizes the cost of materials has a square base of side length 4.45cm and a height of 2.22 cm.
