Download presentation

Published byAbigail Hernandez Modified over 4 years ago

1
**Maxima and Minima in Plane and Solid Figures**

Lesson 8-3

2
Optimization Finding the maximum/minimum (as in the previous lesson) is an important part of problem solving whether in relation to maximizing profit, minimizing cost in manufacturing, of maximizing volume (to mention a few applications). The process of maximizing or minimizing is called optimization.

3
**Optimization Guidelines**

Read and understand the problem. Identify the given quantities and those you must find. Sketch a diagram and label it appropriately, introducing variables for unknown quantities. Decide which quantity is to be optimized and express this quantity as a function f of one or more other variables.

4
**Optimization Guidelines…**

Using available information, express f as a function of just one variable. Determine the domain of f and draw its graph. Find the global extrema of f, considering any critical points and endpoints. Convert the results obtained on step 6 back into the context of the original problem. Be sure you have answered the question originally asked.

5
Example 1: An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting congruent squares from each corner and then bending up the sides. Find the size of a corner square that will produce an open-top box with the largest possible volume.

6
**Example 1: Step 2, 3 and 4 V = LWH V = (16 – 2x)(21 – 2x)(x)**

Domain of V is 0 < x < 8

7
**Example 1: Step 5 Graphically, V = 4x3 – 74x2 + 336x**

Domain of V is 0≤ x ≤ 8 Graphically, Window x[0, 9] y[0, 500] yscl 100

8
**Don't forget to check the endpoints!**

Example 1: Step 6 Recall, critical numbers exist where the derivative is zero or does not exist!!!! So, and Outside the domain! Don't forget to check the endpoints! x = 0 or 8 gives no volume

9
**Example 1: Step 7 Answer the original question!!! x = 3**

V(3) = 4(3)3 – 74(3) (3) = 450 21 – 2x Answer the original question!!! x The volume is maximized at 450 in3 when the corner square is 3 in. x 3 in.

10
Example 2: Find the radius and height of the right-circular cylinder of largest volume that can be inscribed in a right-circular cone with radius 6 in. and height 10 in. 10” r h Step 1 Read and understand the problem 6” Step 2 Draw and label a diagram.

11
**Example 2: Step 3 and 4 So, Step 3 Quantity to be optimized. V = πr2h**

6 Step 4 Express V as a function of one variable. Use similar triangles to get h in terms of r. r 10” h h 10” 10–h 6” So, Note, had we put r in terms of h we would have had to square it.

12
**Example 2: Step 5 Step 5 Determine the domain and graph.**

The radius of the cylinder can not be greater than the cone…6 10” Domain of V is 0 < r < 6 r h 6”

13
**Example 2: Step 6 and 7 So, or Recall,**

Recall, critical numbers exist where the derivative is zero or does not exist!!!! So, or Recall, Therefore, the inscribed cone of largest volume has a radius of 4 in. and height of 3 1/3 in.

14
Example 3: A rectangle is inscribed between the graphs of y = ¼ x4 -1 and y = 4-x2. Find the width of the rectangle that has the largest area. Step 1 Read and understand the problem (x2, y1) (x1, y1) Step 2 Draw and label a diagram. (x1, y2)

15
Example 3: Step 3 and 4 Area = L • W or (x2, y1) (x1, y1) (x1, y2)

16
**Using solve on the TI-89 yields**

Example 3: Step 5 and 6 Domain: Using solve on the TI-89 yields Critical #'s

17
Example 3: Step 7 Therefore, the width of 2(1.064) or about will yield the largest area of the rectangle between the curves. Ω

Similar presentations

Presentation is loading. Please wait....

OK

Optimization Problems

Optimization Problems

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google