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 = πr2h6
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