Optimization: find stamping pattern that minimizes waste

[bodega15]

I have been given this problem as a project and have been working on it for about a week and a half with no progress. I have no idea where to begin. Everytime I think I have something, it's actually nothing. The problem is kind of wordy:

Stamping Blanks from a Steel Coil

Allegheny Ludlum Steel Corporation produces steel from scrap metal at two locations in western PA. During the manufacturing process, hot slates of steel weighing 10 tons are pressed between a series of rollers into thin strips over 1000 feet long and between 15 and 48 inches wide. The strips are called coils because they are rolled up like very long, narrow carpet while they are still hot.

One of the Allegheny Ludlum customers regularly order large quantities of circular steel discs, called blanks, to be used in manufacturing kitchenware. The blanks, 12 inches in diameter, are stamped out of a coil. [The possible patterns are shown in the figures on the left-hand side of the page].)

You are a summer employee at the Brackenridge plant. Your boss, having just learned that you aced your calculus course, has asked you to find a waste-minimizing pattern for stamping out blanks in order to double check the company’s calculations. To keep you honest he has refused to tell you the pattern presently in use. You may simplify the problem as follows:

• Ignore the waste at each end of the coil
• Consider only coils between 15 and 24 inches wide.
• Consider only repeating patterns of the type shown, where the angle "theta" will depend on the width of the coil.

1. Consider a rectangular portion of the coil representing one period of the pattern. Explain why the total area occupied by blanks, or portions of blanks, within such a rectangle is independent of "theta".

2. Express the area of the rectangle in terms of "theta". (Hint: You will need one formula for the situation at left and a different one for the last figure!)

3. Find the value of "theta" that minimizes waste.

Writing assignment:
Write a report to your boss describing the optimal pattern and explaining how you determined that it minimizes waste. The boss took calculus years ago and remembers how to take derivatives. He was never very good at optimization problems so provide plenty of detail.

(Note: You can see the graphics in the PDF document here.)

You need a formula for theta < or = 30 and one for theta > 30 and that's about as far as I got.

 

[stapel]

You must have accomplished something in a week and a half. At the very least, you should have been able to do some computations, finding the different arrangements of blanks for the different widths of coils.

If the coil is fifteen inches wide, how do the blanks fit within the space? What value do you get for theta? What about when the coil is sixteen inches wide?

Looking at the pictured rectangles, and using the widths and their values of theta, what conjectures have you made?

What areas have you obtained? What relationship have you found, if any, between theta, the width, and the area?

Please be specific. Thank you.

Eliz.

 

[galactus]

I'll get you started. Give it a try and write back if you're stymied.


Part 1:

For any vlaue of theta there will be the equivalent of one blank tangent to the left edge of the coil and one tangent to the right edge of the coil within each "period" of the pattern. The total area occupied by the blanks will be \(\displaystyle 2{\pi}(6)^{2}=72{\pi}\;\ in^{2}\)

For part 2: HINT: You will need one formula for \(\displaystyle {\theta}\leq\frac{\pi}{6}\) and one for \(\displaystyle {\theta}>\frac{\pi}{6}\)

The width of the rectangle is \(\displaystyle 12+12cos({\theta}).\) The height is 12 if \(\displaystyle {\theta}\leq\frac{\pi}{6}\), and is \(\displaystyle 24sin({\theta}\) if \(\displaystyle {\theta}>\frac{\pi}{6}\). Now find the area.