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Modeling and Optimization

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Modeling and Optimization

Section 4.4a

To optimize something means to maximize or minimize

some aspect of it…

Strategy for Solving Max-Min Problems

1. Understand the Problem. Read the problem carefully.

Identify the information you need to solve the problem.

2. Develop a Mathematical Model of the Problem. Draw

pictures and label the parts that are important to the problem. Introduce a variable to represent the quantity to be maximized minimized. Using that variable, write a function whose extreme value gives the information sought.

3. Graph the Function. Find the domain of the function.

Determine what values of the variable make sense in the

problem.

To optimize something means to maximize or minimize

some aspect of it…

Strategy for Solving Max-Min Problems

4. Identify the Critical Points and Endpoints. Find where

thederivative is zero or fails to exist.

5. Solve the Mathematical Model. If unsure of the result,

support or confirm your solution with another method.

6. Interpret the Solution. Translate your mathematical result into the problem setting and decide whether the result makes sense.

Guided Practice

An open-top box is to be made by cutting congruent squares of

side length x from the corners of a 20- by 25-inch sheet of tin and

bending up the sides. How large should the squares be to make

the box hold as much as possible? What is the resulting max.

volume?

First, sketch a diagram…

by by

Dimensions of the box?

Volume of the box?

Domain restriction?

Guided Practice

An open-top box is to be made by cutting congruent squares of

side length x from the corners of a 20- by 25-inch sheet of tin and

bending up the sides. How large should the squares be to make

the box hold as much as possible? What is the resulting max.

volume?

Graphical solution?

Graph , calc. the maximum

Analytic solution?

CP:

Check Endpoints and Critical Points:

Guided Practice

An open-top box is to be made by cutting congruent squares of

side length x from the corners of a 20- by 25-inch sheet of tin and

bending up the sides. How large should the squares be to make

the box hold as much as possible? What is the resulting max.

volume?

Interpretation?

Cutting out squares that are roughly 3.681 inches on a side yields a maximum volume of approximately 820.528 cubic inches.

Guided Practice

You have been asked to design a one-liter oil can shaped like

a right circular cylinder. What dimensions will use the least

material?

Sketch a diagram…

3

Volume (in cm ):

2

Surface Area (in cm ):

We need to find a radius and height that will minimize the

surface area, all while satisfying the constraint given by

the volume…

Guided Practice

You have been asked to design a one-liter oil can shaped like

a right circular cylinder. What dimensions will use the least

material?

Guided Practice

You have been asked to design a one-liter oil can shaped like

a right circular cylinder. What dimensions will use the least

material?

A is differentiable, with r > 0, on an interval with no endpoints.

Check the first derivative:

So, what happens

at this r-value?

Guided Practice

You have been asked to design a one-liter oil can shaped like

a right circular cylinder. What dimensions will use the least

material?

Check the second derivative:

This function is positive for all

positive r-values, so the graph

of A is concave up for the entire

domain…

Which means our r-value

signifies an abs. min.

Guided Practice

You have been asked to design a one-liter oil can shaped like

a right circular cylinder. What dimensions will use the least

material?

Solve for height:

Interpretation:

The one-liter can that uses the least material has height equal to the diameter, with a radius of approximately 5.419 cm and a height of approximately 10.839 cm.

Guided Practice

A rectangle has its base on the x-axis and its upper two vertices

on the parabola . What is the largest area the

rectangle can have, and what are its dimensions?

Model the situation:

Dimensions of the rectangle:

by

Area of the rectangle:

Domain:

Guided Practice

A rectangle has its base on the x-axis and its upper two vertices

on the parabola . What is the largest area the

rectangle can have, and what are its dimensions?

Maximize analytically:

Model the situation:

CP:

This corresponds to a

maximum area (…Why?)

Largest possible area is A(2) = 32,

dimensions are 4 by 8.

Guided Practice

2

A 216-m rectangular pea patch is to be enclosed by a fence

and divided into two equal parts by another fence parallel to

one of the sides. What dimensions for the outer rectangle will

require the smallest total length of fence? How much fence

will be needed?

Area of the patch:

Model the situation:

Total length of the fence:

Guided Practice

2

A 216-m rectangular pea patch is to be enclosed by a fence

and divided into two equal parts by another fence parallel to

one of the sides. What dimensions for the outer rectangle will

require the smallest total length of fence? How much fence

will be needed?

Maximize analytically:

The pea patch will measure 12 m by 18 m (with a 12-m divider), and the total amount of fence needed is 72 m.

CP:

This corresponds to a

minimum length (…Why?)