Optimization: A Volume Example
Main Concept

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.

Steps


1.

Identify what value is to be maximized or minimized.

2.

Define the constraints.

3.

Draw a sketch or a diagram of the problem.

4.

Identify the quantity that can be adjusted, called the variable, and give it a name, such as h.

5.

Write down a function expressing the value to be optimized in terms of h.

6.

Differentiate the equation with respect to h.

7.

Set the equation to 0 and solve for .

8.

Check the value of the function at the end points.





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.

Numerical solution





Volume of the box is given by:






First derivative must be found to find a x value that minimizes T



Set the derivative to 0









As h = 7.886751347 is outside the limit, only h = 2.113248653 and the end points should be tested




Hence when h ≈ 2.11cm a maximum volume can be achieved.








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