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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