Algebra 6.1 - Absolute Value
Summary
TLDRIn this educational video, Matt Disorbo explores the concept of absolute value in the context of the Challenger space shuttle disaster. He explains how the cold temperature on launch day compromised the O-rings, leading to the tragic explosion. The video uses a simple mathematical model to illustrate the relationship between temperature extremes and potential damage, emphasizing the importance of considering both hot and cold conditions in safety assessments. It also critiques NASA's data visualization, suggesting simpler charts could have prevented the tragedy.
Takeaways
- đ The script discusses the Challenger space flight disaster, which occurred on January 28, 1986, killing all seven crew members on board.
- đĄ The launch took place in unusually cold weather for Florida, at 28 degrees Fahrenheit, which affected the performance of the O-rings.
- đ O-rings are critical rubber seals in the shuttle that prevent fuel leaks; their failure due to the cold led to the explosion.
- đĄ The script introduces a model using absolute value to predict damage to the O-rings based on temperature extremes, both hot and cold.
- đ NASA had a damage index for O-rings, with 27 or higher indicating dangerous conditions; the model helps predict when this threshold might be exceeded.
- đ The model is represented graphically, showing a 'V' shape where damage increases as temperatures move away from the ideal 70 degrees Fahrenheit.
- đą The script provides a method to solve for temperatures that would result in a damage index of 27, finding critical temperatures of 43 and 97 degrees Fahrenheit.
- â The Challenger disaster was partly due to a focus on high temperatures instead of recognizing the risk of extreme temperatures, including the cold.
- đ The aftermath led to the grounding of the space fleet for nearly three years and highlighted the importance of safety and risk management.
- đ The script critiques NASA's data visualization and presents an alternative by Edward Tufte, which simplifies the presentation for clearer understanding.
- đ€ The video ends with a call to reflect on the presented model and data, questioning its perfection and usefulness in decision-making processes.
Q & A
What was the main topic of the video script?
-The main topic of the video script was the discussion of absolute value functions in the context of the Challenger space flight disaster.
What was the date of the Challenger space flight disaster?
-The Challenger space flight disaster occurred on January 28, 1986.
How many crew members were on board the Challenger during the disaster?
-There were seven crew members on board the Challenger during the disaster, including a teacher.
What was the cost of building the Challenger space shuttle?
-The Challenger space shuttle cost 3.2 billion dollars to build.
What was the role of the O-rings in the Challenger disaster?
-The O-rings, which are rubber rings that seal the joints of the shuttle together, failed due to the unusually cold temperature on the day of the launch, leading to a fuel leak and the subsequent explosion.
What is an absolute value function and how does it relate to the Challenger disaster?
-An absolute value function measures the magnitude of a number without considering its sign. In the context of the Challenger disaster, it was used to model the potential damage to the O-rings from both extremely high and low temperatures.
What was the room temperature at which the O-ring parts were built and assumed to be strong?
-The O-ring parts were built at room temperature, which was assumed to be 70 degrees Fahrenheit.
What was the temperature on the day of the Challenger launch and why was it significant?
-The temperature on the day of the Challenger launch was 28 degrees Fahrenheit, which was unusually cold. This cold temperature caused the O-rings to become brittle and fail, leading to the disaster.
What is the damage index and how was it used in the video script?
-The damage index is a point system used by NASA to track the damage to the O-rings during launches. In the video script, it was used to model the potential damage from both hot and cold temperatures, with an index of 27 or higher considered dangerous.
What was the criticism by Edward Tufte regarding NASA's data visualization of the O-ring damage and temperature?
-Edward Tufte criticized NASA's data visualization as difficult to read and understand due to its complexity. He suggested a simpler chart that plotted temperature on the x-axis and O-ring damage on the y-axis for better clarity.
What is the significance of the Challenger disaster in current safety and risk management discussions?
-The Challenger disaster is frequently used as a case study in safety and risk management to illustrate the importance of considering all potential risks and the consequences of overlooking them.
Outlines
đ Introduction to the Challenger Space Flight Tragedy
This paragraph introduces the topic of the video script, which is the Challenger space flight tragedy. It sets the stage by mentioning the launch date, January 28, 1986, and the presence of a teacher among the seven crew members. The paragraph describes the catastrophic event where the Challenger broke apart 73 seconds into the flight, resulting in the loss of the entire crew. It also touches on the emotional impact of the tragedy, as many children were watching the launch live due to the teacher's involvement. The cost of the Challenger and its successful flights are mentioned, leading into the main question of how the tragedy occurred, and inviting viewers to follow along with the lesson.
đ Investigating the Cause of the Challenger Disaster
This paragraph delves into the technical aspects of the Challenger disaster, focusing on the role of the O-rings, which are critical components in sealing the joints of the shuttle. The script explains that the unusually cold weather on the day of the launch affected the O-rings' performance, leading to a fuel leak and subsequent explosion. The paragraph also discusses the testing of the O-rings and the concept of 'one directional thinking' that may have contributed to NASA's decision to proceed with the launch despite the risks. It introduces a graph that correlates outside temperature with the O-ring damage index, which is used to analyze the potential for damage during launches at different temperatures.
đ Modeling Temperature Impact on O-ring Damage
The script introduces a model to predict O-ring damage based on temperature, using a simple formula that calculates damage as the difference from a 'normal' temperature of 70 degrees Fahrenheit. The paragraph explains how the model works for both hot and cold temperatures, using absolute value to represent the magnitude of temperature differences from the norm. It provides examples of how the model predicts damage at various temperatures, including the critical temperature of 28 degrees Fahrenheit on the day of the launch, which resulted in a high damage index. The paragraph also discusses solving absolute value equations to determine the safe temperature range for launches and the implications of the model for the Challenger disaster.
đ Absolute Value Functions and Their Role in Risk Management
This paragraph continues the discussion on absolute value functions, emphasizing their importance in modeling the potential damage to O-rings from extreme temperatures, both hot and cold. The script presents a graphical representation of the model, showing a 'V' shaped graph that illustrates how damage increases with temperature extremes. It also discusses the identification of safe and dangerous temperature ranges for shuttle launches based on the model. The paragraph concludes with a reflection on the Challenger disaster, highlighting the consequences of not considering both high and low temperatures as risks and the impact of the tragedy on NASA's operations and safety protocols.
đ Data Visualization and Lessons Learned from the Challenger
The final paragraph of the script discusses the importance of data visualization in understanding and communicating risks effectively. It contrasts NASA's original charts, which were criticized for being cluttered and difficult to read, with a simpler alternative presented by Edward Tufte. The paragraph invites viewers to consider whether simplicity in data visualization is indeed better and to reflect on the adequacy and usefulness of the theoretical model presented alongside the actual data. It concludes with a prompt for viewers to consider their own decision-making based on the data and to think critically about the lessons learned from the Challenger disaster.
Mindmap
Keywords
đĄAbsolute Value
đĄChallenger Space Flight
đĄO-rings
đĄTemperature Extremes
đĄDamage Index
đĄRisk Management
đĄData Visualization
đĄModeling
đĄDirectional Thinking
đĄMemorial
Highlights
Introduction to the concept of absolute value in the context of the Challenger space flight disaster.
The Challenger shuttle broke apart 73 seconds into the flight, resulting in the tragic loss of all seven crew members.
The cost of building the Challenger was $3.2 billion, highlighting the financial impact of the disaster.
Explanation of the role of O-rings in sealing the joints of the shuttle and their failure due to cold temperatures.
The use of a damage index to quantify the damage to O-rings during launches, with a value of 27 or higher considered dangerous.
The establishment of a simple model to predict O-ring damage based on temperature, with zero damage at 70 degrees Fahrenheit.
The model's application to predict damage on hot days, showing an increase in damage as temperature rises.
Introduction of absolute value to model damage from both high and low temperatures, emphasizing the magnitude of temperature differences.
Demonstration of solving absolute value equations to find temperatures that would result in a dangerous damage index.
Graphical representation of the absolute value function to model potential damage on both hot and cold days.
Identification of the safe temperature range for shuttle launches based on the damage index model.
Analysis of the launch day temperature of 28 degrees Fahrenheit and its catastrophic impact on the O-rings.
Discussion on the aftermath of the Challenger disaster, including the grounding of the space fleet and the establishment of a presidential commission.
Critique of NASA's data visualization by Edward Tufte, advocating for simplicity and clarity in presenting information.
Comparison of NASA's original chart with Tufte's simplified version, emphasizing the importance of effective data presentation.
Reflection on the question of whether the launch should have been stopped based on the presented data and models.
Final thoughts on the usefulness and limitations of the theoretical model in predicting O-ring damage.
Transcripts
hello mathematicians my name is matt
disorbo covering the allisborough
lessons for
skew the script today we will be
discussing absolute value in the context
of the ill-fated
challenger space flight without further
ado
let's skew it
[Music]
welcome in to lesson 6.1 of our algebra
skew the script series today discussing
absolute value functions specifically
we'll be talking about the challenger
on january 28 1986 the challenger took
off from the coast of florida with a
crew of seven aboard including a teacher
uh here is a photograph of the
challenger's final crew
very unfortunately 73 seconds into the
flight the shuttle broke
apart and sadly none of the crew
survived
in addition many children were watching
live because of the teacher on board
making this an extremely traumatic
experience for america
challenger was very costly to build it
cost 3.2
billion dollars and successfully logged
nine flights and completed 62 days in
space
so the question remains and that's
today's key analysis how did the
challenger tragedy
happen if you'd like to follow along
with our lesson check out the link below
feel free to print or download the
guided notes and work through them as we
talk through the video
to begin our discussion we'll be talking
about absolute value
so on the day of the launch in florida
generally these shuttles are launched
from florida because it's a warm climate
state
but on this specific day the morning was
actually unusually cold 28 degrees
fahrenheit
there uh it was ice on the launch pad
which you can see in the picture in the
left
nasa did decide to proceed anyways
um so to understand more about the crash
we have to understand
o-rings which are rubber rings that seal
the joints of the shuttle together you
can see
um them in the diagram on the right
here's some common examples of o-rings
this is something that you'll see in a
sink or a faucet
obviously for the rocket they are much
much larger
um on the morning of the launch again it
was very cold and the cold temperature
actually broke the seal of an
o-ring which in turn leaked fuel and the
leaked fuel actually caused the
explosion
these two photographs uh here in the
photograph on the left you can see the
red circle
black smoke at the start of the launch
from the o-ring that failed and on the
right
obviously the catastrophic explosion
so like everything else in the shuttle
the o-rings were meticulously tested
so how could nasa make this mistake
um there were many reasons among uh
many many reasons that this happened but
one among several was
one directional thinking um so let's
visually explore the problem of the
o-ring and the challenger with the scrap
here
graph here on the x-axis is outside
temperature and the uh
y-axis is damage index so the o-ring
damage index
it's a point system used by nasa tracks
the damage to the o-ring during launches
a larger index means uh more damage and
we slightly modified this for the lesson
you can see the
original index scale and the link on the
right side
so let's say damage of index of 27 or
higher is dangerous that's our kind of
danger threshold
most of these engineer o-ring parts are
built at room temperature about 70
degrees fahrenheit
so we can assume that they are strong at
this temp so we'll assume zero damage at
the launch with an outside temperature
of 70 degrees fahrenheit
rocket fuel of course is hot so nasa
worries about heat
um so for example when things are hot
o-rings may expand and not seal properly
on a launch pad during a hot day
so combined with the heat created by a
launch hot days could be extremely
dangerous
uh we can actually build a model for the
potential damage on hot days
uh in this case we'll say predicted
damage equals temperature
minus 70. so we can put in our table of
temperature and damage
70 for our temperature um
and if we plug this into our very simple
model we get predicted damage 70 minus
70
or zero so on a date with 70 degrees
fahrenheit
um the damage is expected to be zero
which makes sense because uh
the the this is the temperature of the
rings were built at and you can see
we've we've dotted that out here
what about if we have uh 80 degrees an
80 degree temperature if we plug that
into our simple model
and again we can just use x for
our outside temperature to be more
general um we plug in 80 for x we get 80
minus 70
so predicted damage of 10 and we can
plot that on our chart as you can see
here
um similarly we continue with 90 degrees
our damage becomes 20
plot it in our chart 100 degrees becomes
a damage of 30
and we can draw our nice line connecting
these dots here
um as you would expect as the heat gets
more extreme the potential for damage
rises
so nasa may not launch on extremely hot
days because they'll be
in excess of our our threshold our
damage index
however again on the launch day the uh
opposite actually happened it was
unusually cold 28 degrees fahrenheit
so we if we return to our model what
happens at extremely cold temperatures
um this is where absolute value can be
useful because it turns numbers into
positives we use it when we care about
the magnitude
of a difference so in this case what's
the difference between 86 degrees
fahrenheit and 70 degrees fahrenheit
it's 16 degrees fahrenheit um 86 degrees
is 86 degrees fahrenheit is 16 degrees
higher than
normal or we define normal as 70 degrees
fahrenheit or the room temperature
essentially
how about the difference between 54
degrees fahrenheit and 70 degrees
fahrenheit
well in this case you have a difference
of negative 16 degrees
fahrenheit so 54 degrees is 16 degrees
lower than normal um so
an example if we can turn to this
natural example how different
are 86 degrees fahrenheit and 70 degrees
fahrenheit this is a question about
distance not direction we don't care
about if 86 is higher or lower
we only care about the size of the
difference essentially what we're saying
is what is the magnitude or size of the
difference
in this case it's 16 degrees fahrenheit
86
is 16 degrees away from normal we don't
care about the direction
extending the example how different are
54 and 70 degrees
we see that 54 minus 70 is negative 16
degrees fahrenheit
which is also 16 degrees away from
normal
even though it's lower than 70 whereas
86 is greater so
absolute value measures the magnitudes
of differences
you can see these little vertical lines
we put here these are absolute value
marks
for our notation we get the absolute
value of 16 on the left
absolute value of negative 16 on the
right and what absolute value
essentially does
is it makes it positive so 16 is already
positive it just
stays 16 negative 16 we make it positive
and it turns into
16. so negative 16 turns into 16. um
this shows us that the sizes or the
magnitudes of the differences
are the same so 54 and 86 are both
equally far
from 70. now let's turn to actually
solving for absolute values
again we'll return to our simple model
where predicted damage is x minus 70
x being the outside temperature um again
we saw that if temperature
temperatures are much higher than 70
degrees they're predicted to cause
large damage so nasa will not launch on
extremely hot days
however temperatures that are much lower
than 70 degrees could also be dangerous
o-rings could become brittle in the cold
and break
so how can we model uh predicted damage
from both extremely high
and extremely high extremely low
temperatures both extremely high and
extremely low um in this case we can get
the magnitude
which we can use the absolute value for
of the difference from a normal
70 degree fahrenheit day so our
difference of x the outside temp
from a normal 70 degree day um so here
we can
use our model and actually test it we
have a very hot day of 110 degrees
fahrenheit
we plug it into our model 110 minus 70
we get the absolute value of 40.
40 is already positive so we get a
predicted damage of 40 very high
predicted damage so our model works for
a very hot day
how about a freezing cold day of 30
degrees fahrenheit
we plug in 30 minus 70 the absolute
value of negative 40. remember absolute
value makes things positive we get a
predicted damage of 40.
also very high predicted damage
um we can discuss more specifically
solving inside the actual absolute value
marks so remember that an o-ring damage
index value of 27 is considered
dangerous
so what temperature would create a
predict predicted damage index
of 27. again a predicted damage is what
we're modeling for
in our model so we can set these two
sides equal 27 equals the absolute value
of x minus 70.
um how do we actually solve for that
temperature x because we have these kind
of pesky absolute value markers on the
right side of our equation
remember that the size or magnitude of
the difference between x and 70
has to be equal to 27. so
basically we're saying what numbers have
the same magnitude
as 27 that's going to be positive 27
or negative 27 if we get these two
values
we absolute value them we're going to
get 27.
so we're basically saying that x minus
70 must equal
either 27 or negative 27. so we can have
a positive and negative version of the
side without the absolute value
and that actually allows us to uh write
out the equation
27 equals x minus 70 and negative 27
equals x minus 70. and again we just see
where x minus 70 equals these two values
so simply we can just solve both
equations
um we'll look at both of these equations
in tandem or
separately we'll start with the equation
on the left add 70 to both sides
we get the right side cancels out the
left side becomes 97
equals x um the second equation we add
70 to both sides
right side cancels out left side becomes
43 equals x
and this essentially says we are we
successfully solved our absolute value
equation
this says that temperatures of 97
degrees or or
43 degrees would lead to a prediction
predicted dangerous damage index of 27.
so
uh uh damage index dangerous enough to
not
launch therefore our temperature should
be between
these two temperatures for a safe launch
um turning to an absolute value function
um
again continuing our theme of modeling
potential damage on hot days
here we have predicted damage equals x
minus 70. we have
our chart from earlier um and we're
going to just replace uh
y uh with our predicted damage with y
and
in this case we want to model extreme
days not necessarily just hot days but
hot
or cold days so we have these absolute
value bars around x
minus 70. we can have our handy dandy
table
and plugging in uh 70 for x again we get
y equals absolute value of 70 minus 70
y equals the absolute value of zero
which is zero so our damage is still
zero
which agrees with our point from earlier
if we plug in 80 minus 70
that is the absolute value of 10 which
comes out to 10
damage of 10 agrees with our point from
earlier so so far so good
um we're going to get the same results
we plug in 90. absolute value of 20 is
20
same point from earlier and the same
result if we plug in 100
absolute value of 100 minus 70 is the
absolute value of 30
which is just 30 and we get our same
result familiar so for the hot days
our model works just as well but what
about for cold days
so let's plug in 70 to our value or to
our model again we get absolute value 70
minus 70 that's zero
zero damage on 70 degree days what about
60 days
if we plug in 60 minus 70 we get the
absolute value of negative 10
which comes out to 10. remember absolute
value makes negative numbers
positive so damage of 10. if we plug in
50 we get absolute value of 50 minus 70
absolute value of negative 20 that comes
out to 20
so damage of 20 and finally if you plug
in 40 we get 40 minus 70
absolute value of negative 30 comes out
to 30.
so now we have our full model for y
equals the absolute value of x
minus 70. and you can kind of see the
shape it
takes basically a v shape um
and this happens because the absolute
value absolute value keeps things
positive it keeps the y
axis positive y is positive as x
decreases it's also positive as
x increases and it moves um
higher and higher as x either decreases
or it's actually there
as x increases more um and again we see
that more predicted damage
uh more damage is predicted at extreme
temperatures in either direction it
doesn't matter if it's hot
or cold the more extreme we get the
higher damage is predicted
um now we can actually turn to finding
the temperature boundaries graphically
so again 27 is our dangerous damage
index if we extend that
line we can look at where our graph
crosses we draw our cold point and our
hot point there
extend those down to the x axis and we
see 43
and 97 are where it crosses the the axis
which actually agrees with our model
from earlier
and in between there we have our safe
temperature zone where the damage index
is below
27. however remember that on the day of
the launch
it was 28 degrees fahrenheit
if we put that on our graph on launch
day you can see the 28 over there
we can draw a vertical line extend our
damage function and we see that we get a
pretty high
damage when we have 28 degrees
fahrenheit we can actually solve
for what the damage would be we use our
model y equals absolute value of x minus
70.
plug in 28 for x 28 minus 70. get the
absolute value of negative 42
which of course just becomes 42. so a
damage of 42 when it's 28 degrees
outside that's far far above the danger
well in excess of the danger boundary of
27
um for our damage index so the result
again
uh very unfortunately was tragedy uh
partly because engineers are worried
about high temperatures instead of just
extreme temperatures hot or cold
the launch proceeded the o-rings became
so brittle uh from the cold they failed
to seal
and the result was uh catastrophic
uh the aftermath um was definitely very
challenging the space place the space
fleet was grounded for
nearly three years it was a presidential
commission established by president
reagan to investigate nasa
and this uh challenger is now frequently
used as a case study
for safety and risk management and you
can see the memorial for the challenger
crew
in arlington on the right um finally
we're going to turn to data
visualization and we'll actually
uh dovetail this into our discussion so
during testing and prior launches nasa
actually did chart the o-ring damage in
the outside temperature
at launch you can see it here on the
left however uh data visualization
expert edward tufte
uh very famously criticized these charts
as difficult to read and understand
and i will let you decide for yourself
if that chart on the left
is very easy to comprehend or not um
tufti uh gave a uh alternative to this
chart
which just plots the temperature on the
x-axis first the o-ring damage on the
y-axis
definitely much simpler and looks like
sort of the chart we were working on
um we've created here with the same
chart and data but with using the damage
scale that we actually
created in this lesson again much easier
to to visualize
so um if you had this data would you
have stopped the launch
and you want to justify your answer
using the data that you've seen here
um tufty was famous for arguing that
these data visuals should be simple
and avoid drawings figures and other
chart junk you can see how tough to easy
example
was much simpler than nasa's which had
sort of a lot of
diagrams and numbers all over the place
and it's more difficult to read
and our second discussion question is do
you agree with tuffy that simpler is
better
and explain your reasoning and address
these two charts tupt's chart
versus nasa's charter finally you can
look at our theoretical model alongside
the data
and ask yourself is our model perfect is
it useful
explain why or why not that's all for
today on skew the script thanks for
coming and we'll see you next time
[Music]
Voir Plus de Vidéos Connexes
Space Shuttle Challenger Disaster: Major Malfunction | Retro Report | The New York Times
Hot and Cold - Heat | Class 7 Science Chapter 3 | CBSE 2024-25
Bagaimana Proses Terbentuknya Embun? #BelajardiRumah
Disaster Readiness & Risk Reduction ( DRRR)- M1-Lesson 2: HAZARD, EXPOSURE, VULNERABILITY & CAPACITY
Introduction to Thermal Physics
Heat | Grade 8 Science DepEd MELC Quarter 1 Module 4 Part 1
5.0 / 5 (0 votes)