# G12ES project_2012

```ATHS&amp;IAT
G12 ES Term I
Math Project
This project covers the application of the early
topics in calculus ( Differentiation and its applications )
G12 ES_MATH Project
2
Term 1
(2012/13)
Date due: Nov 14
12 ES
SECTION:
Students’ Name:
Differentiation Project
In this project you will work with one of the most important processes of
calculus Differentiation. You will apply the rules you have learnt to find
answers of real life problems that have faced the aircraft engineers during
a full Airbus journey.
Derivatives
Basic Rules
Implicit Diff &amp;
Related Rates
Applications of
Differentiation
G12 ES_MATH Project
3
G12 ES_MATH Project
4
Internet Search
Introducing Airbus A 380, Collect the following information about that
aircraft.


Airbus A380 characteristics.

How long it takes an A 380 to takeoff.

What is the takeoff velocity?

What is the takeoff distance, i.e. the runway length that will be
crossed before it takes off?
Waiting at the airport
The average number of airplanes waiting in a modern airport to take off can
b e m o d e l e d b y t h e f u n c t i o n 𝑤(𝑥) =
𝑥2
2(1−𝑥)
where 𝑥 is a quantity between 0 and 1
known as the traffic intensity. Find the rate of change of the number of
a i r p l a n e s w a i t i n g w i t h r e s p e c t t o t h e t r a f f i c i n t e n s i t y w h e n 𝑥 = 0.1 a n d 𝑥 = 0.7
G12 ES_MATH Project
5
Taking-off Distance and Time
Suppose that the distance an Airbus travels along a runway before tak ing
off is given by
10
𝐷 (𝑡 ) = ( ) 𝑡 2
9
meter measured from the starting point, and 𝑡 is
measured in seconds from the time the brakes are released. The aircraft
will become airborne when its speed reaches 72 meters per second.
How long will it take to become airborne, and what distance will it travel in
that time?
G12 ES_MATH Project
6
Speed of an airplane by Radar
If this airplane was flying on a flight path that would take it directly over
a radar station, as shown in figure below. If “s” was decreasing at a rate of
400 miles per hour when “s=10miles” what is the speed of the plane?
Aircraft Glide Path
Now the Airbus starts descent from an altitude of 1 mile, 4 miles west of
the runway.
F i n d t h e 𝑎, 𝑏, 𝑐 a n d 𝑑 i n t h e f u n c t i o n
[−4,0]
𝑓(𝑥) = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
on the interval
that describes a smooth glide path for the landing.
The function you founded models the glide path of the plane. When would
the plane be descending at the greatest
rate?
G12 ES_MATH Project
7
Shockwave
http://goo.gl/cwYeo or http://goo.gl/E9Svc
Now if the frequency F of an airbus landing on runway heard by a
stationary observer on 𝐹 =
𝑣+𝑣𝐿
𝑓
𝑣+𝑣𝑠 𝑠
where F is frequency perceived by the
o b s e r v e r , 𝑓𝑠 i s t h e f r e q u e n c y o f t h e s o u r c e , 𝑣 i s t h e v e l o c i t y o f t h e w a v e s , 𝑣𝐿
i s t h e v e l o c i t y o f t h e l i s t e n e r a n d 𝑣𝑠 t h e v e l o c i t y o f t h e s o u r c e .
F i n d a n e q u a t i o n o f t h e 𝐹 i n t e r m o f 𝑣𝑠 . T h e n f i n d t h e r a t e o f c h a n g e o f 𝐹
w i t h r e s p e c t t o 𝑣𝑠 , I n t w o c a s e s o n c e w h e n a i r b u s i s a p p r o a c h i n g a t v e l o c i t y
of 200km/hr, and the other whe n its moving away at a velocity 200km/hr
Presentation
Prepare a presentation including a wiki to show your findings, include
videos, your own work, and other related information.
Produce a QR code for your wiki.
G12 ES_MATH Project
8
Projects Rubric
Guidelines
Criteria
4
Completeness
20%
Presentation
and
Integration of
Technology
70%
Creativity&amp;
enrichment
10%
Points
3
2
1
totally
completed
and correct.
(100%)
completed,
OR
Partially
wrong.(75%)
partially
completed,
AND
Partially
wrong
(50%).
Attempted
(25% or
less)
____
Students used one
mean of
technology. The
tool used helped
the student and
was useful to
support his project.
Moreover, the
student was able to
explain the work
he/she submitted
confidently and
fluently; he/she
was able to
colleagues and
instructor’s
questions
Student used a
mean of
technology but it
was not that
supportive to the
student was able to
explain the work
he/she submitted
confidently and
fluently and he/she
reflected an
understanding of
his/her works. The
student was able to
colleagues and
instructor’s
questions.
Student was
able to explain
the work he/she
submitted.
Student
reflected a
shallow
understanding
of his/her
work; she was
some of
colleagues and
instructor’s
questions,
Student use of
technology was
primitive and
way below the
level of other
IAT students.
Student was
unable to explain
the work he/she
submitted.
Student reflected
no understanding
of his/her work;
he/she was
unable to
colleagues and
instructor’s
questions.
____
outstanding
aspects of his/her
project.
outstanding
aspects of his/her
project.
outstanding
very few
aspects of
his/her project.