Section 14.2 Application of Extrema

RevenueThe sale of compact disks of “lesser” performers is very sensitive to price. If a CD manufacturer charges p dollars per CD, where p = 12 – q/8 , then q thousand CDs will be sold.

a) Find the revenue function R(q).b) Find the number of CDs must be sold to maximize

the revenue.c) Find the maximum revenue and price per CD that

produces the maximum revenue.

Optimization Problem

1. Assign variables and draw a figure/diagram, if possible.

4. Use the conditions of the problem to write the quantity to be optimized as a function of ONE variable.

5. Find the feasible domain of that function

6. Find the absolute extrema of the function on the specified domain.

2. Write the condition/constrain for the situation

3. Write an equation that relates the quantity to be optimized to other variables.

Area

Find the dimensions of the rectangular field of maximum area that can be made from 300 m of fencing material.

125 m

25 m

100 m

50 m

75 m

75 m140 m

10 m

Packing Design

A television manufacturing firm needs to design an open-topped box with a square base. The box must hold 32 cu. in. Find the dimensions of a box that can be built with the minimum amount of materials.

An open box is to be made by cutting a square from each corner of a 12-in. by 12-in piece of metal and then folding up the sides. What size square should be cut from each corner to produce a box of maximum volume?

Volume

A fence must be build to enclose a rectangular area of 20,000 sq. ft. Fencing material costs $2.50 per foot for the two sides facing north and south, and $3.20 per foot for the other two sides. Find the cost of the least expensive fence.

Minimum Cost

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?

x x

40 2x

40 2A x x

240 2A x x

40 4A x

0 40 4x

4 40x

10x 40 2l x

w x 10 ftw

20 ftl

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?

x x

40 2x

40 2A x x

240 2A x x

40 4A x

0 40 4x

4 40x

10x

10 40 2 10A

10 20A

2200 ftA40 2l x

w x 10 ftw

20 ftl