- anonymous

can anybody help me??
1) Simon’s cell phone plan charges a $16 monthly
service fee plus $0.04 per minute. Megan’s plan
charges $0.12 per minute with no monthly fee.
How many minutes do each of them have to use
to pay the same amount?
1A) what the intersection represents
1B) coordinates of intersection

- chestercat

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- anonymous

Hello, I would love to help you with this problem. Have you started this problem already or do you have an idea of what the steps should be?

- anonymous

hi....I'm all lost... :( ...please help me from the first

- anonymous

alrighty then. I think the first step is to set up a system of equations. Have you studied this before?

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## More answers

- anonymous

:( .....noooooo .... need to start with x and y ?????

- anonymous

Well that is what we will do then. Why don't we make the first equation with x and y look something like this: \[.04x+16=y\] this equation tell us how much Simon will spend each month for his cellphone. "Y" tells us what his total monthly charge is. The .04x is part of the fee where "X" is how many minutes he uses each month and the 16 is the $16 fee he is charged automatically. Does this make sense?

- anonymous

ohh.... is it? .....and like this I can say.:
.12x = y......right?

- anonymous

you there?

- anonymous

:(

- anonymous

Yes that is right. So your second equation is going to be \[.12x=y\] and that is going to tell us how much megan's bill will be. where again "x" is the number of minutes she used that month and "y" is the bill total.

- anonymous

but I wonder what would be the next??

- anonymous

Are you righting this down? Because we are about to go back to the original question with this new information and it will be helpful to keep track of our steps :)

- anonymous

yeah.....I'm all set to go ahead... :)

- anonymous

okay so look at the two equations you have on your paper. Do you see what they have in common? They both use the same variables, which are "x" and "y".

- anonymous

yeah

- anonymous

Well the first part of the question is asking us for how many minutes they must use to have the same bill. So we know that if "x" is the number of minutes they use and "y" is their bill, these values must be the same for both equations right?

- anonymous

ohh....yeah...

- anonymous

okay well why don't we set the two equations equal to each other? What I mean by equal to each other is use what they have in common:
\[.04x+16=y | y=.12x\] see where you could set them equal to each other? try putting the two together like this:
\[.04x+16=.12x\]
they have the same answer for "y" so what we can do is take advantage of this by setting the two equations to be "equal" to each other.

- anonymous

right... :) ....and I"ve solved it....the value of x should be 200

- anonymous

but what about the next parts??? :(

- anonymous

YES! So if x=200 and thats the number of minutes they used then what we really have is 1/2 the co-ordinates to graph this problem. Why don't we take our newfound value of "x" and plug it into each equation separately? so we can have:
\[.04(200)+16=y \] and
\[.12(200)=y \] and then solve for y in each one of the equations to get Simon and Megan's bill totals.

- anonymous

and y = 24

- anonymous

yup it is! Now you have x and y values for both equations. But they are the same... how does this help? Well look at the question 1A). What does the intersection represent? Well consider that if 2 equations have the same point in common they are intersecting. What this means for us is that at the point of intersection, megan and simon used the same number of minutes and had the same bill. (remember that they both used 200 minutes and both of them had the same bill amount of $24.00)

- anonymous

I'm trying to understand all...please be with me

- anonymous

its alright take your time ;) is there a specific part you're having trouble with?

- anonymous

you mean the point of intersection would be (x,y) or (200, 24) ??

- anonymous

THAT IS CORRECT! If you were to draw a graph of the functions on the x,y plane they would intersect at (200,24) and what that means or represents is that when they have both used 200 minutes their phone bills will be the same.

- anonymous

okkkkkkkkkkkkkkk...........that's great.........thanks a lottttttttttt ... :)

- anonymous

can you help me with one more problem???

- anonymous

yup I sure can :)

- anonymous

:) ....so kind of you......

- anonymous

look at the sequence of numbers in the set: {7^1, 7^2, 7^3, 7^4.... 7^1998}
How many of the numbers in the set have a ones digit of 9?

- anonymous

Hahaha I'm really really really bad with sequences but I will give it a shot. {7^1=7,7^2=49,7^3=343,7^4=2401,7^5=16807...7^1998=a number so big it won't show up on most calculators but mine says it ends in 9....} I think we need to find a pattern to the system. So our first interaction with a 9 in the one's place is at the value of 7 squared or 7^2. May I ask what class is this for? I am gonna guess an algebra course or something similar? ;)

- anonymous

:( .....I"m in grade eight ....I've got hint like ... There are 1,998 numbers in the set. And the terms are:
7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807
7^6 = 117649
7^7 = 823543
7^8 = 5764801
...
The ones digit is the very last number. Note that the ones digit has a repeating pattern: 7, 9, 3, 1, 7, 9, 3, 1, ..., 9
and the pattern is four terms long.

- anonymous

Now... The ones digit is a 9 when the exponent of 7 is (4n-2), where n is a positive integer:

- anonymous

and n would be 2,6,10,......but...I need to solve what could be the last value of n for which the ones digit will be 9....... and I'm lost

- anonymous

:(

- anonymous

okay lets see....

- anonymous

to start I think I would take note of the fact that the 9 shows up in the one's place every "4" different exponent expansions... So with that prediction lets guess and see if 7^10 will end in a 9 (or have a 9 in the one's place). The answer I received with my calculator program is: 282475249 which does end with a 9 in the ones place. So if every 4 terms a 9 will show up we can calculate the total number.
the equation they give you of (4n-2) is right because our first encounter with a 9 in the ones place happened at 7^2 which is only 2 terms into the sequence. Thats where the 2 in (4n-2) comes from. They didn't tell you what to set (4n-2) equal to directly but we are supposed to set it equal to 1998. :)
\[4n-2=1998\]
or more accurately
\[4n=2000\] since we will move 2 over by adding it :)

- anonymous

but is the 1998th term has a 9 in the one's place?

- anonymous

so the question is asking whether or not 7^1998 has a 9 in the ones place?

- anonymous

no no...I"m asking it....does it contain a 9 in the one's place? or what is the nearest number of 1998 containing9 in one's place?

- anonymous

ahhhh okay gimme just a second I think I've got it now ;)

- anonymous

okay I am going to go back to your original question. "look at the sequence of numbers in the set: {7^1, 7^2, 7^3, 7^4.... 7^1998} How many of the numbers in the set have a ones digit of 9? " is that the way the question actually looks or what it says? If not what does the question state? If its really looking for a number containing a 9 in the ones place that follows the 7^(4n-2) pattern then it would be 49...

- anonymous

no....it's asking how many numbers are there in the set: {7^1, 7^2, 7^3, 7^4.... 7^1998} containing 9 in the one's place

- anonymous

oh well then we kinda already answered that ;) that was using the equations we were looking at earlier. 4n-2=1998. If there are 1998 numbers in this set. meaning exponents of 7 from 1 to 1998 how many of them will have a 9 in the ones place. Well lets follow our equation. 4n <- because every 4th term will have a 9 in the ones place. and then -2 <- because the first time we see a 9 in the sequence is the 2nd term. So our equation states that the 2nd term and every fourth term after that will have a 9 in the ones place. Then 1998 is the last exponent we are going to evaluate. So when we solve 4n-2=1998 for n we will see that there are 500 numbers in this set that will end with a 9 in the ones place.

- anonymous

that means that there are 500 numbers where 7^some number will have a value where 9 is in the one's place

- anonymous

ohhh.......ok....now got it.....

- anonymous

yeah sorry for the confusion. I am really bad with number sets and patterns. :) Do you have any other questions?

- anonymous

ohh no....it helped me and you are really good........ :) ...but yeah....one last question.... :(

- anonymous

haha thank you for the compliment. And what is the question?

- anonymous

How many solutions does 3 cos2 x = 1 have on the interval [0, 2π)

- anonymous

I saw that question float by earlier. I'll give it a shot too ;)

- anonymous

okay what do you know about the interval of [0,2pi]?

- anonymous

0 to 360 degree interval

- anonymous

correct. Have you ever done any graphing using [0,2pi) before? Solving this problem graphically is the quickest and easiest way but there are other methods if you have not studied graphing of sine and cosine functions.

- anonymous

please explain me first and then I"ll try to do it graphically by myself... :)

- anonymous

you there??.... :(

- anonymous

haha okay. Well the basic graph of a cosine wave oscillates (moves back and forth) vertically from -1 and 1 on a graph. This means that the y-value of the graph is going to change from -1 to 1 over and over again. Have a look at this graphic I made to explain what it is. http://i625.photobucket.com/albums/tt338/npsgaming/Math%20Help%20On%20OPEN%20STUDY/CosineWave.jpg?t=1301026735

- anonymous

yeah...that's clear to me

- anonymous

So looking at that graph you can see the y value is changing from -1 to 1 as it moves along the x-axis. On the x-axis you will see pi and 2pi labeled. This will help us in a moment. With this visual in mind, think about the interval of [0,2pi) it includes the value 0 and and doesn't include 2pi but everything between there is a valid answer.

- anonymous

So we will use this graph again when it comes time to plot our function 3cos(2x)=1. The interval limits our solutions to anytime the line crosses the x-axis. Here is another image that is the graph of 3cos(2x)=1. http://i625.photobucket.com/albums/tt338/npsgaming/Math%20Help%20On%20OPEN%20STUDY/3cos2xeq1.jpg?t=1301027056 that is what the graph of 3cos(2x)=1 looks like. Now take not of where 0 and 2pi are. How many times do you see the line crossing the x-axis? (its the red lines :) )

- anonymous

6 times

- anonymous

thats how many times are shown, but remember the interval of 0 to 2pi limits our solutions to only the lines that are between 0 and 2pi. So while there are 6 lines shown, only 4 are within the interval of [0,2pi)

- anonymous

hmmmm.....but how can I do it without graph??

- anonymous

have you ever used the unit circle to answer a problem? (thats where the 360 degrees you mentioned earlier comes from)

- anonymous

not sure....can you show me the steps so that I can have an idea???

- anonymous

:(

- anonymous

haha I will do my best. I was never any good at geometry either ;)

- anonymous

does the problem ask you for the specific solutions? or is it just asking for how many? :)

- anonymous

cuz I can show you using the unit circle and triangle how many solutions there are :)

- anonymous

it's just asking how many solutions

- anonymous

okay

- anonymous

this link will show you the unit circle. Notice that its radius is 1. and the unit circle is broken up into quadrants just like a graph. There are 4 quadrants. 0-90deg, 90-180deg,180-270deg, and 270-360deg. http://www.processing.org/learning/trig/imgs/unit_circle.jpg
Does this make sense so far? (this is a pretty difficult topic and I'm not sure I even knew what the unit circle was until 9th grade or so.

- anonymous

yeah ...got it....but what about the problem???? ... :(

- anonymous

okay so our problem 3cos(2x)=1 has 4 possible solutions. Because our interval is from 0 to 2pi this includes all 4 quadrants of the unit circle.... Each quadrant has an equivalent version of the answer. Right now im working on trying to get the exact answers.... bear with me ;) Its been a few years since I touched the unit circle ;)

- anonymous

:).... you have already helped me so much.........I'll be happy to wait.

- anonymous

have you ever heard of the arc cosine? (sometimes written as:
\[\cos^{-1} \]

- anonymous

yeah..

- anonymous

okay I think I may have it then ;)

- anonymous

one solution is \[1/2 \cos^-1(1/3)\] that is for quadrant 1 and you can get that by using some algebra. 3cos(2x)=1
into cos(2x)=1 but from there I'm no sure where to go :(

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