The height of the container should be equal to 2.57 cm, so as to minimize cost.
<h3>How to calculate the height?</h3>
First of all, we would determine the volume of the rectangular container by using this formula:
Volume = l × w × h
Where:
Substituting the given parameters into the formula, we have;
Volume = x × 4x × h
Volume = 4x²h
At a volume of 12 ft³, we have:
12 = 4x²h
h = 12/4x²
h = 3/x²
For the total area of top and bottom, we have:
Area = (x × 4x) + (x × 4x)
Area = 4x² + 4x²
Area = 8x² sq. units.
Thus, the cost of top and bottom is given by:
Cost = 8x² × 1.50
Cost = $12x²
For the total area of all four sides, we have:
Area = (x × h) + (x × h) + (4x × h) + (4x × h)
Area = xh + xh + 4xh + 4xh
Area = 10xh sq. units.
Thus, the cost of all four sides is given by:
Cost = 10xh × 4.50
Cost = $45xh.
Also, the cost function would be given by:
C(x) = 12x² + 10xh
C(x) = 12x² + 10x(3/x²)
C(x) = 12x² + 30/x
Taking the first derivative, we have:
dC/dx = 24x - 30/x²
24x = 30/x²
24x³ = 30
x³ = 30/24
x = 1.08
Therefore, the height will be:
h = 3/x²
h = 3/1.08²
h = 3/1.1664
Height, h = 2.57 cm.
Read more on minimum height here: brainly.com/question/27308167
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