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Quote of the Day:
"Bees … by virtue of a certain geometrical forethought …
know that the hexagon is greater than the square and the
triangle, and will hold more honey for the same
expenditure of material." -- Pappas
Objectives:
The student will learn to solve applied maximum minimum
problems.
1. Collect Homework.
2. Method for Solving Applied Max/Min Problems (overhead)
1. Determine what is to be maximized or minimized.
2. Write an algebraic relationship defining the
quantity being maximized or minimized.
3. Write any other appropriate algebraic
relationships.
4. Express the quantity to be maximized or
minimized as an equation in one variable
5. Sketch a graph of the above relationship with
particular emphasis on the domain of the
function.
6. Determine critical values in the domain to
determine likely places for maxima and
minima.
7. Determine the maximum or minimum value
of the function over the appropriate domain.
Be sure to check for absolute maxima and
minima, not just relative maxima and minima.
3. Examples:
Example #1 (On overhead)
Determine the maximum product of two positive numbers
whose sum is 8.
Solution:
1. Maximize the product.
2. Product = ___ x ____
3. Sum = ___ + ___ = 8
4. Product = x (8 – x)
5. Graph Product vs x
6. dP/dx = 8 – 2x ==> x = 4
7. x = 4, y = 4
8. Product = (4)(4) = 16.
Example #2
Determine the maximum area of a rectangle that can
be enclosed with 100 meters of fence.
Show different possible pens for your pet:
30x20 25x25 10x40 1x49 (call S.P.C.A.)
let L = length and W = width of rectangle
Then A = L * W
P = 2L + 2W = 100
So, L + W = 50
L = 50 – W
A = (50 – W) * W
A = 50W – w^2
dA/dW = 50 – 2W
0 = 50 – 2W (why?)
W = 25 meters
L = 25 meters
Area = 225 square meters
How does this change if you need only to fence in 3
sides (if you build the pen against one side of the
house)?
4. Optimization
Six engineers and six mathematicians are attending a conference and are traveling by train. One by one, each of the engineers goes up to the ticket counter and buys a ticket to the conference. But only one of the mathematicians does. The engineers look puzzled and one of the mathematicians says, "Optimization." The twelve get on the same car and one mathematician stands at each end of the car. Now the engineers are really puzzled. After a while, the mathematician at one end, yells, "Conductor!" On that cue, all the mathematicians pile into the rest room and lock the door. The conductor enters the car and announces, "Tickets, please. Tickets!" He passes the engineers and punches each of their tickets. At the end of the car, he notices the restroom is occupied and knocks on the door, "Ticket, please." The ticket slides out from under the door, he punches it and slides it back, then leaves the car and continues to the next car. The engineers look at each other and decide how clever the mathematicians have been, and then wink at each other. They all attend the conference and have a good time. Upon arriving at the train station, one engineer buys a ticket and they giggle at each other. The mathematicians do not buy any. This time again, the engineers look puzzled, and the same mathematician says, "Optimization." This time all the mathematicians sit down and the engineers have the lookouts. One engineer, peers down a couple of cars and shouts, "Conductor!" Immediately all the engineers pile into the rest room, while the mathematicians just sit there. Once the engineers are in the rest room, one of the mathematicians knocks on the door and says, "Ticket, please." The ticket slides out under the door, the mathematician grabs it and along with the other mathematicians, runs to the other rest room and they lock themselves in.
5. Assignment: Worksheet on Max/Min Problems -- Do #1, 2, 3, 4, 8 |