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Hi! Its round two of scribe posts... Today we began class with the POD. The problem was: Without using a calculator or your snowman sheet, find an exact value for sec π/6.
Mac went up to the board and wrote this:
She explained that she knew that π/6 was equal to 30º, so she could use the 30º,60º,90º triangle to find out the answer for the question. She also knew that Sec = 1/cos, so if cos=x/r, then sec=r/x.
All 30º,60º,90º triangles proportionally have a hypotenuse equal to 2, a long leg equal to 3, and a short leg equal to 1. So now, with the information she had above, she could figure out that a) r= 2 b) x= √3
So, sec= 2/√3
Sec = 2/√3 x √3/√3 = 2√3/3
so, the final answer is : sec = 2√3 /3
The last thing we saw yesterday was a theorem that stated:
If r < sn =" T1">
Mac went up to the board and wrote this:
She explained that she knew that π/6 was equal to 30º, so she could use the 30º,60º,90º triangle to find out the answer for the question. She also knew that Sec = 1/cos, so if cos=x/r, then sec=r/x.
All 30º,60º,90º triangles proportionally have a hypotenuse equal to 2, a long leg equal to 3, and a short leg equal to 1. So now, with the information she had above, she could figure out that a) r= 2 b) x= √3
So, sec= 2/√3
Sec = 2/√3 x √3/√3 = 2√3/3
so, the final answer is : sec = 2√3 /3
The last thing we saw yesterday was a theorem that stated:
If r < sn =" T1">
With the information this theorem gave us we were then asked by Mr. Alcantara to find the answer for the following problem:
A ball is dropped from 100 ft and bounces straight up. On each bounce the ball climbs to half of it’s previous height. Assume the ball bounces forever. How far will the ball travel?
After several minutes trying to figure out the problem Mr. A gave us a series of steps we could follow in order to find our way through the problem easier.
1st: List the terms
2nd: Find out, is the series Arithmetic or Geometric? To do this we can use the tests we already know, divide the 2nd term by the 1st term, and then the third by the 2nd to see if they are the same, or subtract the second term from the third term, and the first term from the second one. 3rd: Is the series infinite or finite?
4th: Is r <>
The answers for the questions for this problem are:
1) 100, 100, 50, 25,
2) Geometric, r = ½
3) Infinite
4) Yes.
Now, we are ready to use the equation we have.
Sn = T1 /1 – r
Sn = 100 + 100 /1 – ½
Because 100 is both term one and term 2 we need to eliminate one of them when finding Sn, and then add it at the end, so that it is actually a series.
Sn = 100 + 200 ft Sn = 300 ft
FLASHBACK: When we did this exact problem but with only ten bounces the distance covered by the ball was 299.8 ft, so the ball only traveled .2 feet in infinite-10!
When we were finished working through this problem Daniel Rubio asked a question that most of us were asking to ourselves: “What happens if r > 1?
Mr. A answered this question by making us realize that when r is grater than 1 the numbers will get bigger and bigger, so it would be impossible to calculate infinity.
Then we were presented with a new problem, what happens if the ball bounces to 2/3 of its previous height?
SdC went to the board, where she wrote:
D = 100 + 100 = 100 + 100 + 3 = 100+300 = 400 ft
1/3 1 1
We then realized, that there was a problem with SdC answer, because 100 isn’t T1… T1 is 100(4/3), because it is actually 2 (100(2/3)) = 100(4/3)100(2/3) is how high the ball bounces, but, it travels that distance when it goes up, and when it goes down, so we have to multiply the distance by 2.
With T1 settled, we could now plug it in to the equation
D = 100 + 2 * 100(2/3) /(1/3)
D = 100 + (400/3)
D = 500 ft
Mac then made a comment, she thought that her way of looking at the problem was actually easier. What she does, is separate the problem into two series, one series is the distance the ball goes up, and the other series is the distance that the ball goes down.
So, her problem would look like this:
Distance Down:
Sn = 100/1-(2/3) = 1007(1/3) = 300 ft
Distance Up:
Sn = 100 (2/3)/1 - (2/3) = 66.6/(1/3) = 200 ft
Now, she adds up the tow distances, and gets 500ft, the same answer we got using the other method!
Moving on from this problem, Mr. Alcantara touched a subject a lot of math students spend a lot of time thinking, “When will I use this?” “How does this apply to real life?”
He showed us an example right there, in the classroom. He jiggled the video beam, and it moved, a great distance at first, and slowly the distance got smaller and smaller and smaller, just like a geometric series.
He then connected this to constructions that need to be earthquake resistant and told us how engineers use this idea so that buildings will move with the earthquake and not destroy.
We then moved into a new section of this topic.
There are two types of Infinite Series (adding an infinite numbers of numbers).
Mr. A answered this question by making us realize that when r is grater than 1 the numbers will get bigger and bigger, so it would be impossible to calculate infinity.
Then we were presented with a new problem, what happens if the ball bounces to 2/3 of its previous height?
SdC went to the board, where she wrote:
D = 100 + 100 = 100 + 100 + 3 = 100+300 = 400 ft
1/3 1 1
We then realized, that there was a problem with SdC answer, because 100 isn’t T1… T1 is 100(4/3), because it is actually 2 (100(2/3)) = 100(4/3)100(2/3) is how high the ball bounces, but, it travels that distance when it goes up, and when it goes down, so we have to multiply the distance by 2.
With T1 settled, we could now plug it in to the equation
D = 100 + 2 * 100(2/3) /(1/3)
D = 100 + (400/3)
D = 500 ft
Mac then made a comment, she thought that her way of looking at the problem was actually easier. What she does, is separate the problem into two series, one series is the distance the ball goes up, and the other series is the distance that the ball goes down.
So, her problem would look like this:
Distance Down:
Sn = 100/1-(2/3) = 1007(1/3) = 300 ft
Distance Up:
Sn = 100 (2/3)/1 - (2/3) = 66.6/(1/3) = 200 ft
Now, she adds up the tow distances, and gets 500ft, the same answer we got using the other method!
Moving on from this problem, Mr. Alcantara touched a subject a lot of math students spend a lot of time thinking, “When will I use this?” “How does this apply to real life?”
He showed us an example right there, in the classroom. He jiggled the video beam, and it moved, a great distance at first, and slowly the distance got smaller and smaller and smaller, just like a geometric series.
He then connected this to constructions that need to be earthquake resistant and told us how engineers use this idea so that buildings will move with the earthquake and not destroy.
We then moved into a new section of this topic.
There are two types of Infinite Series (adding an infinite numbers of numbers).
Type 1: Infinite Arithmetic Series
Chewy and Bee chose two random numbers, 5 and 11
If 5 = T1 and 11 = r
T1 = 5
T2 = 16
T3 = 27
With that string of numbers we realize that the numbers will get bigger and bigger as they approach infinity, which makes it impossible to calculate.
In this cases Sn = DNE as it approaches to infinity.
When d > 0, Tn increases without bound. So, Sn = DNE as it approaches infinity.
When < sn =" DNE">
T1 = 5
T2 = 16
T3 = 27
With that string of numbers we realize that the numbers will get bigger and bigger as they approach infinity, which makes it impossible to calculate.
In this cases Sn = DNE as it approaches to infinity.
When d > 0, Tn increases without bound. So, Sn = DNE as it approaches infinity.
When < sn =" DNE">
Type 2: Infinite Geometric Series
In this cases, when:
· r = 0, so Tn will always be equal to 0, because 0 times any number is 0. If this is true, then Sn will always be equal to T1
· r = 1, then Tn never changes, because any number times one is always that number. So, when T1 is not equal to 0, Sn will be equal to infinity or negative infinity, depending on what T1 is. This means, that, Sn=DNE
· r = -1, then Tn oscillates between two numbers. This means that Sn = DNE
We finished talking about this, and the bell rang, bringing our class to an end.
The next scribes will be Gaby and Chewy!
Homework: The homework on the board was: PG.500-503, PG.502, #1-19 ODD
