Transcript Optimization - UTEP MATHEMATICS

Visual Calculus
Ana Franco
Amra Kanlic
Gabriel Mendoza
Alister Ng
Alfredo Rodriguez
Lina Yick
University of Texas at El Paso
April 6,2002
Introduction
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Intervention - Cone problem
Video- Classroom activity
Pretest
Posttest
Results
Acknowledgements
Discussion
Intervention
On a hot, sunny El Paso day, you and your buddies decide to buy
some ice cream. When you get to the ice cream shop, there’s a sign
that reads,
"SALE! Eat ice cream by making your own waffle cone for only
$1.00!" .
The store sells you a circular waffle and allows you to roll it up into a
cone, which will be filled with ice cream for a $1. Of course you want
the most amount of ice cream in your waffle cone.
Finding Equation
8 cm
diameter
8
8
8
8
r
h
r 2 + h2 = 8 2
"
h = ± 64 - r 2
8
8
h
r
V=
1
(1)
(2)
(3)
h p r2
3
Substitute the value of h into equation (3).
V Hr L =
1
3
pr
2
"
64 - r 2
(4)
Graph of diameter v.s volume
d y
p
d2
Vi
=
d 2 $ 64 k2 {
12
4
V (ml)
250
225
200
175
150
125
100
75
50
25
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
d (cm)
Students’ Data
S#
V (ml)
1
4
260
220
23
21
200
180
2
25
22 8
14
20 13 24
19
11 17
12
240
26
7
6
15
5
18
16
3
9
160
10
11
12
13
14
d (cm)
15
16
17
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
D (cm)
V (ml)
14
14
14
13.9
15
12.25
12
14
10
15.5
12
12.5
12
14
13.5
14.5
12.5
14.5
11.5
11.5
11
13.5
11
12.5
14
12
265
250
205
265
235
205
215
242
180
157.5
235
235
235
235
210
215
235
215
240
240
205
243
205
235
250
220
Students’ graph
V (ml)
250
19 20
225
200
9
175
22
4
1
2
25
8
14
5
12
13 17
11 26 24 15
18
7
23
16
3
21
6
10
150
125
100
75
50
25
1
2
3
4
5
6
7
8
9
d (cm)
10 11
12
13
14 15
16
Calculus Solution
Take the derivative with the respect to r.
V' Hr L =
p i
3
"
2 r
64 - r 2 -
k
r3
è
y
64 - r 2 {
p i 2 r H64 - r 2 L - r 3 y
=
è
3 k
64 - r 2
{
p i - r H3 r 2 - 128 L y
=
è
3 k
64 - r 2
{
V' HrL= 0, 64 - r2 š 0 andr š 0.
To solve
Thus,
3 r 2 - 128 = 0
128
r = ±$
=±8$
3
r = 8 $
2
3
2
ª 6.532 cm
3
d = 2 r ª 13.064 cm
i
V
k
8 $
2 y
3 {
ª 206.37
cm3 = 206.37
ml
Pretest
Three firefighters are called about a fire at a house (H),
figure 1. The fire station is located at F. Each firefighter
Ana, Bobby, and Roberto leave the fire station with an
empty bucket. They each pick a path to the ocean to fill
their bucket with water. Then they proceed to the burning
house.
Figure 1. Three firefighters, at station F, first run to ocean
for water and then run to the burning house. Notice that
the first letter in their name marks the path to the ocean
of each firefighter.
Pretest Graph
Figure 2. The distance of each firefighter is marked on
the graph by the first letter of their name.
Questions
a)
b)
c)
d)
e)
f)
Who ran the shortest distance to the ocean?
By person, rank the distance they had to run to the ocean
(from shortest to longest).
1.
2.
3.
Who ran the shortest total distance to the house?
By person, rank the total distance they ran from the fire
station to the burning house (from shortest to longest).
1.
2.
3.
Is there an optimal way to run to shorten your distance from
the station to the house? Explain.
Which way would you run to the ocean to create the shortest
path from the station to the burning house (create your own
path)? Label your path on figure 1 and also on figure 2.
Explain.
Posttest
Four fighter fighters: Ana(A), Bobby(B), Edith(E) and Roberto (R)
are called about a fire at a house (H), figure 1. The fire station is
located at F. Each firefighter leaves the fire station with an empty
bucket.
Figure 2. The distance of each firefighter is marked on
graph by their first letter of their name.
Questions
a) By using figure 2, who ran the best way to the
house? Why?
b) To save the house from the fire, which way
would you run? If there is not a point on the
graph, then mark your own point and explain
this path on figures 1 and 2.
Pretest Solution
F
H
B
A
X
Ocean
R
Posttest Solution
Results
16
14
12
10
correct answer
almost correct
wrong answer
8
6
4
2
0
pre
post
Acknowledgements
Alister’s 8th grade classes at St. Joseph
Erandi Perez and O. Perez
Veronica Herrera
References
Lott, W. Johnny and Smith, Paul. (1979). Reflections on
putting out a fire. School Science and Mathematics, 79
(5), 434-38.
Sobel A. Max and Maletsky M. Evan. (1999). Teaching
mathematics : a sourcebook of aids, activities, and
strategies. Boston, MA: Allyn and Bacon.
http://www.math.utep.edu/Student/alfredo/opt/optimization.html
Discussion
Any questions?
Thank you.
http://www.math.utep.edu/Student/alfredo/opt/optimization.html