Hi, It’s me, Daniel Rubio and it is my turn to be the scribe. Today’s POD was this one:
“Pier becomes possesed by the devil so that he can spin his head all the way around. Suppose that Pier’s head is a perfect sphere and that he sins his head at a constant rate of 8 revolutions per minute. A small ant is perched on Pier’s nose and in 6 seconds it travels 60 cm. Fin the radius of Pier’s head.”
A small hint: Find the centra angle that the ant sweeps.
As a first step lets make a picture to make understanding easier.
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Now that we have made ourselves clear lets look for ways to get to the solution of the problem. In class, we saw 3 different approaches to the problem: Susy DC’s, MAC’s , and Mr. Alcantara’s approach to the problem. Before startig to explain each way to solve the problem, it is important to know which data we have.
-Pier’s head is spinning at a constant rate of 8 revs/min
-The Fly traveled 60 cm in 6 sec ::: Important note: we can use this as sector length to solve the problem :::
Now that we’ve made that clear, we can start on the solutions.
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Now, moving forward we look into MAC’s approach; which has a similar reasoning but a different execution
This album is powered by BubbleShare - Add to my blog
Now moving on to Mr Alcantara’s approach, which uses a simpler reasoning on the problem
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Now we are FINALLY done with the POD… it took me a while to do those ones.
I am now going to post about today’s ACTUAL lesson.
Sybil the rat : A mathematicla answer .
-Today we reviewed the problem of Sybil the rat : A mathematicla answer .
Sybil must 1st cross half of a room, the ¼ of it, then 1/8 and so on.
First thing we need to realize is, tn = fraction of room to cross. So when we are in doubt and don’t know what to do, we list the first terms of the sequence, which in this case are:
½ , ¼ , 1/8 , 1 /16 … as we could realize the explicit definition for this sequence is: n
(1/2)
>So we need to ask ourselves a question… what happens to tn as n approaches infinity?
First, we should give a graphical answer- REMEMBER that your grapgh shoud include
>A scale with labels
> tn values as it approaches its limit ( in this case zero)
> tn values as n approaches infinity
>must label both axes
>must show points not lines( remember it is a sequence)
>Interpret the graph
>Now what if we wanted to know the total amount of room crossed?
Sn= total amount of room crossed
If Sybil can cross the room then Sn> or equal to 1
List find first terms of the series.
Realize that it is a geometric series whichs r = 1/2
Well that was basically the most important part, Mr. Alcantara said something about using algebra on limits to cancel them out and get somenthing which I didn’t really understand. Mr. Alcantara said he did’t expect us to know it, so I’m not covering it now.
Homework was : Read 500-503
Pg. 502 1-11 odd
n
In the book they mention a new theorem, that if l r l < 1, then lim r = 0.
n->infinity
If you think about it it is totally logical because you are repeatedly multiplying by a number less than one, each time oits going to go closer to zero.
Well that’s it for today.
“Pier becomes possesed by the devil so that he can spin his head all the way around. Suppose that Pier’s head is a perfect sphere and that he sins his head at a constant rate of 8 revolutions per minute. A small ant is perched on Pier’s nose and in 6 seconds it travels 60 cm. Fin the radius of Pier’s head.”
A small hint: Find the centra angle that the ant sweeps.
As a first step lets make a picture to make understanding easier.
This album is powered by BubbleShare - Add to my blog
Now that we have made ourselves clear lets look for ways to get to the solution of the problem. In class, we saw 3 different approaches to the problem: Susy DC’s, MAC’s , and Mr. Alcantara’s approach to the problem. Before startig to explain each way to solve the problem, it is important to know which data we have.
-Pier’s head is spinning at a constant rate of 8 revs/min
-The Fly traveled 60 cm in 6 sec ::: Important note: we can use this as sector length to solve the problem :::
Now that we’ve made that clear, we can start on the solutions.
This album is powered by BubbleShare - Add to my blog
Now, moving forward we look into MAC’s approach; which has a similar reasoning but a different execution
This album is powered by BubbleShare - Add to my blog
Now moving on to Mr Alcantara’s approach, which uses a simpler reasoning on the problem
This album is powered by BubbleShare - Add to my blog
Now we are FINALLY done with the POD… it took me a while to do those ones.
I am now going to post about today’s ACTUAL lesson.
Sybil the rat : A mathematicla answer .
-Today we reviewed the problem of Sybil the rat : A mathematicla answer .
Sybil must 1st cross half of a room, the ¼ of it, then 1/8 and so on.
First thing we need to realize is, tn = fraction of room to cross. So when we are in doubt and don’t know what to do, we list the first terms of the sequence, which in this case are:
½ , ¼ , 1/8 , 1 /16 … as we could realize the explicit definition for this sequence is: n
(1/2)
>So we need to ask ourselves a question… what happens to tn as n approaches infinity?
First, we should give a graphical answer- REMEMBER that your grapgh shoud include
>A scale with labels
> tn values as it approaches its limit ( in this case zero)
> tn values as n approaches infinity
>must label both axes
>must show points not lines( remember it is a sequence)
>Interpret the graph
>Now what if we wanted to know the total amount of room crossed?
Sn= total amount of room crossed
If Sybil can cross the room then Sn> or equal to 1
List find first terms of the series.
Realize that it is a geometric series whichs r = 1/2
Well that was basically the most important part, Mr. Alcantara said something about using algebra on limits to cancel them out and get somenthing which I didn’t really understand. Mr. Alcantara said he did’t expect us to know it, so I’m not covering it now.
Homework was : Read 500-503
Pg. 502 1-11 odd
n
In the book they mention a new theorem, that if l r l < 1, then lim r = 0.
n->infinity
If you think about it it is totally logical because you are repeatedly multiplying by a number less than one, each time oits going to go closer to zero.
Well that’s it for today.
1 comment:
PD: THe animal on the tip of pier's nose is an ant, not a fly jeje.
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