Applications of Extrema Lesson 6.2. A Rancher Problem You have 500 feet of fencing for a corral What...
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Transcript of Applications of Extrema Lesson 6.2. A Rancher Problem You have 500 feet of fencing for a corral What...
Applications of Extrema
Lesson 6.2
A Rancher Problem
You have 500 feet of fencing for a corral
What is the best configuration(dimensions) for a rectangularcorral to get the most area
One side of the rectangle already has a fence
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Sample Problem
Your assistant presents you witha contract for signature• Your firm offers to deliver 300 tables to a
dealer at $90 per table and to reduce the price per table on the entire order by $0.25 for each additional table over 300
What should you do?• Find the dollar total involved in largest
(smallest) possible transaction between the manufacturer and the dealer.
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Solution Strategy
1. Read the problem carefully• Make sure you understand what is given• Make sure you see what the unknowns are
From our problem• Given
• 300 tables at $90 per table• $0.25 reduction per table on entire order if > 300
• Unknowns• Largest possible transaction• Smallest possible transaction 4
Solution Strategy
2. If possible sketch a diagram• Label the parts
From our problem• Not much to diagram …• More likely in a problem about the size of a
box to minimize/maximize materials or volume
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x + 3
x2x
Solution Strategy
Decide on a variable to be maximized (minimized)• Express variable as a function of one other
variable• Be sure to find function domain
From our problem• T = transaction amount• T = f(x) = ?
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90 300( )
(90 .25 ) 300
x for xf x
x x for x
Solution Strategy
To analyze the function, place it in Y= screen of calculator
Check the table (♦Y) to evaluate the domain and range for setting the graph window
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Solution Strategy
4. Find the critical points for the function• View on calculator
For our problem
• Use derivative tests to find actual points
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Note jump in discontinuous
function
Note jump in discontinuous
function
300 .25xD x x
Solution Strategy
5. If domain is closed interval• Evaluate at endpoints, critical points• See which value yields absolute max or min
For our problem
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maximummaximum
minimumminimum
Strategy Review
1. Read carefully, find knowns, unknowns
2. Sketch and label diagram
3. Determine variable to be max/min• Express as function of other variable• Determine domain
4. Find critical points
5. If domain is closed interval• Check endpoints• Check critical points 10
Practice Problem
A fence must be built to enclose a rectangular areaof 20,000 ft2
• Fencing material costs $3/ft for the two sides facing north and south
• It costs $6/ft for the other two sides
Find the cost of the least expensive fence
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Assignment
Lesson 6.2
Page 383
Exercises 5 – 33 odd
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