The document discusses applications of maxima and minima problems involving Kepler's problem, minimum ladder length problems, and their mathematical solutions. It presents three problems: 1) Kepler's wine barrel pricing problem based on wet rod length, 2) minimum ladder length through a fence, and 3) extending the latter to a circular fence. The solutions find the volume/length expressions, take derivatives, and set to zero to find maxima/minima. Effective technology use in education requires careful integration, selecting abstract topics, teacher training, and intelligent software choice.

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SUBTOPIC: APPLICATIONS OF
MAXIMA AND MINIMA
PRESENTED BY :
HIMANI ASIJA
P.G.T. MATHEMATICS
DELHI PUBLIC SCHOOL, VASANT KUNJ

2. EFFECTIVENESS OF USE OF
TECHNOLOGY
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For the best results of use of technology in education, it is
required to
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Carefully integrate technology and the content
Select some abstract topics in the subject (math) and use
technology as a medium of instruction
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Focus on professional development of teachers (both
in service and pre service)
Make an intelligent choice on the kind of software to be
used

3. REFLECTIONS ON THE USE OF
0011 0010 TECHNOLOGY FROM AROUND
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THE WORLD
 Is highlighted in the “Principles and Standards of Mathematics”,
U.S.A.
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 Ministry Of Education, France since 2004, for future teachers
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 Ministry Of Education, Singapore from 2006 for senior classes
and from 2007 for primary classes.

4. Use of technology to make teaching of
calculus innovative
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Subtopic: Applications of Maxima and Minima
Problems discussed:
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1. Kepler’s problem.
2. To find the minimum length a ladder through a vertical
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fence of a given height and at a given distance from a wall.
3. Extension of the above problem with a circular fence.

5. Problem 1: Kepler’s problem
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Statement :
In 1612, the great scientist Johannnes Kepler was about to remarry
and he needed to buy wine for the celebration. The wine salesman
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brought barrels of wine to Kepler’s house and explained how they
were priced: a rod was inserted into the barrel and diagonally
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through a small hole in the top. When the rod was removed, the
length of the rod which was wet determined the price for the barrel.

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H L

7. A MATHEMATICAL SOLUTION OF
KEPLER’S PROBLEM
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Length of the rod (L) = fixed, since Kepler wanted to keep his
bugdet fixed.
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Now, volume of the cylinder is given by V= Π R2 H
Also, L2 = R2 + H2
SO, V = Π (L2 - H2 ) H
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On finding the first derivative and equating it to zero,
V´(H) = Π (L2 - 3H2 )
V´(H) = 0
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H= L (Omitting the negative sign as h is positive)
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FIND V˝(H) AND OBSERVE IT TO BE NEGATIVE

8. Using G.S.P. finding the solution of the
problem
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9. Problem 2: Minimum height of ladder
through a fence
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Statement: To find the minimum length of a ladder
passing through a fence of given height at a given distance
from the wall.
WA
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L L
(y) LADDER (L)
F
E
N
a
C
E
b
x

10. A MATHEMATICAL SOLUTION
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Assuming a=3 and b=3.
Using geometry, y 3
x x 3
9x2
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l2 x2
( x 3) 2
On differentiation w.r.t. x, and equating the derivative to zero,
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we obtain x=6 and y=6 and l=6√2
On finding second differential, at x=6, we observe that it is
positive and hence ensures a minimum length of the ladder.

11. Using G.S.P. finding the solution of the
problem
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12. Problem 3 : Extension of the above
problem with a circular fence
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Statement: To find the minimum length of a ladder passing
through a circular fence.
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W LADDER
A
L
L
y
r=3
FENCE
x

13. A MATHEMATICAL SOLUTION
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Given circular fence of radius r
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Minimize length of the ladder BE=L
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Let OB=y and OE=x, then by geometry,
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AB=BC=y-3 and DE=CE=x-3
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O
So, we have L = x+y-6 and L2 = x2 +y2 D E
Giving us L= y 2 6 y 18
y 6
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On finding dL/dy, and putting it =0, y=6 3 2
y= 6+3 2 Satisfies the given condn.(i.e. finding L , it comes out
to be positive, thus ensuring a minima.)

14. Using G.S.P. finding the solution of the
problem
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15. CONCLUSION AND
RECOMMENDATIONS
Technology emphasizes 1011
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rote learning.
Teacher student interaction increases
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To make the use of technology most effective, it is important
To correctly choose the topics to be taught using technology
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To train the present teachers to use the technological aids
To make it an integral part of the pre service teacher training
To change the assessment techniques
That educators and programmers work together
Last, but not the least, to make the teachers believe that technology
is not their replacement, but an aid for teaching.