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Design Problems in Three Dimensions

Optimization of surface area and volume.

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Design Problems in Three Dimensions

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The steps to answer three-dimensional modeled problems:

  1. Create your model.
  2. Test possibilities or create an equation that shows the relationship between the variable you are looking to maximize or minimize and another variable in the problem.
  3. Graph and look for a max or min (in necessary).
Tip: Always make sure you know what you're looking for. Pay attention to the equation you use and the variables involved.
Answer the following questions:
  1. A typical peanut butter jar is a cylinder with a height of 4 inches and a diameter of 3.9 inches. Compare and contrast this type of jar with the new truncated pyramid container. 

    An open faced box is being made from a square piece of paper that measures 10 inches by 10 inches. The box will be made by cutting small congruent \begin{align*}x\end{align*}  by \begin{align*}x\end{align*}  sized squares out of each corner. 

  2. What's an equation that relates \begin{align*}x\end{align*}  to the volume of the box?

  3. Graph the equation from #2 and explain what it shows.

  4. What size squares should you remove from each corner to maximize the volume?

  5. Why does it not make sense to try to minimize the volume of the box?

  6. Come up with at least 5 rectangles with a perimeter of 24 inches. Which rectangle has the biggest area? 

  7. If you did not consider a square in #6, compare the area of a square with perimeter 24 inches to the other rectangles that you came up with. What do you notice?

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