Answer :
The minimized amount of material used to make the box is 13516 cm²
Let the side of the square base be x cm
According to the problem, the volume of the box is 13500 cm³
Therefore,
x² X height = 13500
or, height = 13500/x²
Now it is given that the top is open.
Hence, the box only has one square side.
Hence the surface area of the box will be
x² + 2x.13500/x² + 2x.13500/x²
= x² + 54000/x
Therefore, f(x) = x² + 54000/x
Now, to minimize the above problem, we equate the first derivative to 0
f'(x) = 2x - 54000/x² = 0
or, (2x³ - 54000)/x² = 0
or, 2x³ = 54000
or x³ = 27000
or, x = 30
Now, we check whether the above result is a minimal value or not.
f''(x) = 2 + 108000/x³
or, f"(30) = 2 + 108000/54000
= 4
Since f"(30) is positive, the minimized value of x is 30.
Therefore, the minimal amount of material used is
4² + 54000/4
= 16 + 13500
= 13516 cm²
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