The largest volume possible from one piece of paper for open-top box is 64.296 cubic unit.
<h3>What is meant by the term maxima?</h3>
- The maxima point on the curve will be the highest point within the given range, and the minima point will be the lowest point just on curve.
- Extrema is the product of maxima and minima.
For the given question dimensions of open-top box;
The volume is given by the equation;
V = (8.5-2x)(11-2x)(x)
Simplifying the equation;
V = x(4x² - 39x + 93.5)
Differentiate the equation with respect to x using the product rule.
dV/dx = x(8x -39) + (4x² - 39x + 93.5)
dV/dx = 8x² - 39x + 4x² - 39x + 93.5
dV/dx = 12x² - 72x + 93.5
Put the Derivative equals zero to get the critical point.
12x² - 72x + 93.5 = 0.
Solve using quadratic formula to get the values.
x = 4.1 and x = 1.9
Put each value of x in the volume to get the maximum volume;
V(4.1) = 4.1(4(4.1)² - 39(4.1) + 93.5)
V(4.1) = 3.44 cubic unit.
V(1.9) = 1.9(4(1.9)² - 39(1.9) + 93.5)
V(1.9) = 64.296 cubic unit. (largest volume)
Thus, the largest/maximum volume possible from one piece of paper for open-top box is 64.296 cubic unit.
To know more about the maxima, here
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↔ .5
↔ 2.0
)
↔ 0.86603...
↔ -2.23607...