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More Optimization Problems

# More Optimization Problems - More Optimization Problems...

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More Optimization Problems Because these notes are also being presented on the web we’ve broken the optimization examples up into several sections to keep the load times to a minimum. Do not forget the various methods for verifying that we have the optimal value that we looked at in the previous section. In this section we’ll just use them without acknowledging so make sure you understand them and can use them. So let’s get going on some more examples. Example 1 A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the dimensions of the window be to let in the most light? Solution Okay, let’s ask this question again in slightly easier to understand terms. We want a window in the shape described above to have a maximum area (and hence let in the most light) and have a perimeter of 12 m (because we have 12 m of framing material). Little bit easier to understand in those terms. Here’s a sketch of the window. The height of the rectangular portion is h and because the semicircle is on top we can think of the width of the rectangular portion at 2 r . The perimeter (our constraint) is the lengths of the three sides on the rectangular portion plus half the circumference of a circle of radius r . The area (what we want to maximize) is the area of the rectangle plus half the area of a circle of radius r . Here are the equations we’ll be working with in this example. In this case we’ll solve the constraint for h and plug that into the area equation.

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The first and second derivatives are, We can see that the only critical point is, We can also see that the second derivative is always negative (in fact it’s a constant) and so we can see that the maximum area must occur at this point. So, for the maximum area the semicircle on top must have a radius of 1.6803 and the rectangle must have the dimensions 3.3606 x 1.6803 ( h x 2 r ). Example 2 Determine the area of the largest rectangle that can be inscribed in a circle of radius 4. Solution Huh? This problem is best described with a sketch. Here is what we’re looking for. We want the area of the largest rectangle that we can fit inside a circle and have all of its corners touching the circle.
To do this problem it’s easiest to assume that the circle (and hence the rectangle) is centered at the origin. Doing this we know that the equation of the circle will be and that the right upper corner of the rectangle will have the coordinates . This means that the width of the rectangle will be 2 x and the height of the rectangle will be 2 y . The area of the rectangle will then be, So, we’ve got the function we want to maximize (the area), but what is the constraint? Well since the coordinates of the upper right corner must be on the circle we know that x and y must satisfy the equation of the circle. In other words, the equation of the circle is the constraint. The first thing to do then is to solve the constraint for one of the variables.

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More Optimization Problems - More Optimization Problems...

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