Modeling and Optimization

1. Modeling and Optimization Section 4.4a

2. 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.

3. 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.

4. 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?

5. 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:

6. 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.

7. 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…

8. 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?

9. 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?

10. 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.

11. 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.

12. 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:

13. 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.

14. 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:

15. 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?)