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.

Fill in the Blanks

If you are given 200 feet of fencing with which to construct a rectangular chicken pen, what is the greatest possible area of the pen? (type in numbers only, not units)

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