Using Geometry to Solve Design Problems

Using Geometry to Solve Design Problems

Lesson, Notes, & Practice

Open Ended

If you are given 50 feet of border with which to enclose a rectangular flower bed, sketch at least 3 different rectangles you could construct. Type in the areas of your 3 rectangles.

If we use whole number units, there are many different rectangles we could make. If we used fractional units, there are many more we could make!

How do I maximize area of a rectangle with a given perimeter?

How to optimize area (summary)

1) Write your perimeter equation

2) Simplify by dividing by 2

3) Use substitution method to write your equation with one variable

4) Your equation is a quadratic with a negative leading coefficient!

5) Find the vertex to maximize area using (-b/2a, A(-b/2a))

6) This will tell you ONE of the dimensions (x-coordinate of vertex) and the maximum area (y-coordinate of vertex)

7) Find the other dimension by dividing the area by the dimension you do know.

Open Ended

If you have a square sheet of paper that is 30 cm on each side, you can cut squares out of each corner to fold it up and make an open-topped box. Sketch at least 2 possibilities (or make them with a sheet of paper) and calculate the 2 possible volumes.

Key ideas about solving maximizing volume problems

• The "height" is the side length of the square you cut out from each corner

• Your length will be the original length minus 2 times height (L - 2H)

• Your width will be the original width minus 2 times height (W - 2H)

• Therefore, volume can be modeled by V = H(W-2H)(L-2H)

• Helpful website: https://sites.math.washington.edu/~conroy/general/boxanim/boxAnimation.htm

Multiple Choice

You have a sheet of paper that is 10" x 12". If you cut out the corners to make an open topped box, which height will give you the maximum volume?

• 0.6 inches

• 1.2 inches

• 1.8 inches

• 2.4 inches

There are many other types of problems we can solve using geometric modeling, but these are the 2 "new" types

• maximize area

• maximize volume