Optimal Value of Measurements (Two Dimensions)
Optimization is the process of finding values that make a given quantity the greatest (least) possible given certain conditions.
Maximum – greatest possible
Minimum – least possible
Example 1 (Maximize Area Given Perimeter):
a) Suppose you have 40 m of fencing. You want to make a rectangular dog pen. What is the maximum area you can provide for a dog?
∴ The maximum area we can provide for the dog is 100m2
b) What dimensions give us the maximum area?
The dimensions that give us the maximum area are 10m by 10m
So, the shape that gives us the maximum area is a square.
Example 2: Workers at a resort set up a rectangular area to store outdoor equipment and furniture. They use metal stands. They have 26 stands, each 3 m long. The storage enclosure they set up could have different shapes. How many stands should be used for the width and length to make the largest possibleenclosure?
∴ There should be used & stands along the and 6 stands along the width or vice versa.
OR
Psquare=45
244=6
Example 3: Suppose we have a rectangle with an area of 24 units2 . What dimensions will give us the minimum perimeter? What shape will minimize the perimeter
The shape that provides the minimum perimeter is almost a square.
Conclusion: For a rectangular enclosure with a given perimeter, if fencing is required on all sides, the optimal are occurs when the enclosure is a square
Example 4: You have 80 metres of fencing. What dimensions will give the largest possible area? What is the maximum area?
Solution: Given: P=80m
Aim for a square.
P=45sub. P=80
80=45solve for s
804=454
20=s
∴ s=20
∴ The dimensions that will give us the largest possible area are 20m by 20m
Asquare=S2
=202
=2020
=400
∴ The maximum area is 400m2
