Optimization Problems

Finding maximum and minimum

Example Problem: Product Find two nonnegative numbers whose sum

is 9 and so that the product of one number and the square of the other number is a maximum.

General Guidelines for Optimization 1. Read each problem slowly and carefully.

Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly.

2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts.

3. Define variables to be used and carefully label your picture or diagram with these variables.

This step is very important because it leads directly or indirectly to the creation of mathematical equations.

4. Write down all equations which are related to your problem or diagram. Clearly denote that equation which you are asked to maximize or minimize.

Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint" equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs.

5. Determine the domain of the function and set an appropriate window on the calculator.

6. Use the maximum/minimum function to find the values.7. Interpret your results and answer the question.

Example Problem: Product Find two nonnegative numbers whose sum is 9 and so that

the product of one number and the square of the other number is a maximum. Quantities: two numbers, x and y Optimization Equation:

Trying to maximize product of x and y2: Product=xy2

Constraint Equation: Two numbers whose sum is 9: x+y=9

Substitute the constraint equation into the Optimization Equation:

y=9-x, so Product=x(9-x)2

IMPORTANT: Optimization Equation should contain ONLY ONE variable.

Appropriate Domain: Nonnegative numbers: Domain: [0, 9]

Graph and solve: Answer: 3 and 6, Product = (3)(62)=108

Example Problem: Capacity A manufacturer wants

to design an open box (a top with open top) having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

Optimization Equation: V=x2h

Constraint Equation: x2+4xh=108

Equation to maximize: V=x2[(108-x2)/(4x)] V=27x – x3/4

Example Problem: Capacity

Practice Problem: Distance Which points on the

graph of y=4-x2 are closest to the point (0,2)?

2 2

2

2 22 2 2 2

Optimization Equation:

0 2

Constraint Equation:

4

Equation to minimize:

0 4 2 2

d x y

y x

d x x x x

Domain: ,

Reasonable Window: 0,2

Answer: 1.225, 2.5

Practice Problem: Area A rectangle is

bounded by x-axis and the top half of a circle centered at the origin with radius 5. What length and width should be the rectangle so that its area is a maximum?

22 25

[0,5]

3.536, 25

dimension: 7.071 by 3.536

Area x x

x

x Area

Practice Problem: Cost A cylindrical can is to hold 20 m3 The

material for the top and bottom costs $10/m2 and material for the side costs $8/m2 Find the radius r and height h of the most economical can.

Practice Problem: Time A little duckie is in the ocean swimming at a

location 2.5 miles off the coast. A restaurant is located along the seashore, 4.5 miles down the coast. The duckie can swim to a point on the coast, and then run along the coast to the restaurant. The duckie can swim 1.2 miles/hour, and can run 1.6 miles/hour. If the restaurant is in an urgent need to make a roast duck dish, what is the least amount of time that takes for the duckie to get to the restaurant?

Practice Problem: Ladder Find the length of the

shortest ladder that will reach over an 8-ft high fence to a large wall which is 3 ft behind the fence.

Practice Problem: Angle What angle between two edges of length 3 will

result in an isosceles triangle with the largest area?

Practice Problem: Viewing Angle A movie screen on a

wall is 20 feet in height and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle of the movie screen is as large as possible?

Practice Problem: Total Area Four feet of wire is to

be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area?

Minimize Carbs

Fin…