1. Sect. 3-7 Optimization

2. Definition If something is made as small as possible
it is called minimized
If something is made as large as possible
It is called maximized

3. Optimize
To find the greatest, least, optimum, maximum or minimum of something

4. Problem Solving Strategies
Define variables to represent the quantities in the problem. A sketch may be needed
Identify the quantity to be optimized with a short phrase, such as “maximum area”
Write a primary equation for the quantity to be optimized
Reduce the primary equation to one unknown, this may involve a 2nd equation.
Think about the domain
Determine absolute extrema from extreme value theorem, 1st derivative test, or the 2nd derivative test
Be sure to answer the question, is it reasonable?

5. 1) Find two positive numbers whose sum is 110 and whose product is a maximum

6. 2) A particle moves along the x-axis so that its position is given by Find the minimum velocity of the particle and the time at which it occurs. Justify your answer .

7. 3) An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 20 cm wide and 28 cm long by cutting out a square from each corner and then bending up the sides. Find the size of the corner squares which will produce a box having the largest possible volume.

8. 4) A farmer has 3600 meters of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?

9. 5) A poster is to contain 100 square inches of picture surrounded by a 4-inch margin at the top and bottom and a 2-inch margin on each side. Find the overall dimensions that will minimize total area of the poster.

10. 6) Construct a window in the shape of a semi-circle over a rectangle
6) Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 12 feet, what are the dimensions of the window with maximum area?

11. 7) A rectangle is to be inscribed in a semicircle of radius 2
7) A rectangle is to be inscribed in a semicircle of radius 2. What is the largest area the rectangle can have, and what are the dimensions?

12. Homework
Page 223
# 4, 5, 10, 11,18, 21, 22, 25 and 34
and
Worksheet 3-C