Buffalo Bills Ranch North Platte Nebraska Greg Kelly Hanford High School Richland Washington Photo by Vickie Kelly 1999 A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn What is the maximum area that you can enclose ID: 538396
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Slide1
4.4 Modeling and Optimization
Buffalo Bill’s Ranch, North Platte, Nebraska
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1999Slide2
A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
There must be a local maximum here, since the endpoints are minimums.Slide3
A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?Slide4
To find the maximum (or minimum) value of a function:
1 Write it in terms of
one
variable.
2 Find the first derivative and set it equal to zero.
3 Check the end points if necessary.Slide5
Example 5:
What dimensions for a one liter cylindrical can will use the least amount of material?
We can minimize the material by minimizing the area.
area of
ends
lateral
area
We need another equation that relates
r
and
h
:
Motor
OilSlide6
Example 5:
What dimensions for a one liter cylindrical can will use the least amount of material?
area of
ends
lateral
areaSlide7
If the end points could be the maximum or minimum, you have to check.
Notes:
If the function that you want to optimize has more than one variable, use substitution to rewrite the function.
If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check.
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