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Optimization: A Volume Example
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Main Concept
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An optimization problem involves finding the best solution from all feasible solutions. One is usually solving for the largest or smallest value of a function, such as the shortest distance or the largest volume.
A minimum or maximum of a continuous function over a range must occur either at one of the endpoints of the range, or at a point where the derivative of the function is 0 (and thus the tangent line is horizontal). These are called critical points.
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Steps
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1.
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Identify what value is to be maximized or minimized.
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2.
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Define the constraints.
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3.
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Draw a sketch or a diagram of the problem.
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4.
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Identify the quantity that can be adjusted, called the variable, and give it a name, such as h.
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5.
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Write down a function expressing the value to be optimized in terms of h.
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6.
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Differentiate the equation with respect to h.
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7.
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Set the equation to 0 and solve for  .
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8.
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Check the value of the function at the end points.
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Problem: Alice is given a piece of cardboard that is 20cm by 10cm. She wants to make an open top box by cutting the corners and folding up the sides.
Let h be the height of the box. Adjust the value of h using the slider to find the value that maximizes the volume.
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Numerical solution
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Volume of the box is given by:
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First derivative must be found to find a x value that minimizes T
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Set the derivative to 0
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As h = 7.886751347 is outside the limit, only h = 2.113248653 and the end points should be tested
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Hence when h ≈ 2.11cm a maximum volume can be achieved.
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