CPhill

domain and range of {(-2,4), (3,9), (-3,4), (5,1)}

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In a simplistic manner, "domain" asks, "What "x" values are included in the set?"

These would be {-2, 3, -3, 5}

Likewise, "range" asks, "What "y" values are included in the set?"

These would be {4, 9, 1}

Thanks, Bertue.........you must have heard the old story about Gauss using EXACTLY that method to find the sum of the 1st 100 digits as an classroom assignment when he was 10 years old.

I much prefer your method to mine!!!

why is arccos of -3.225 invalid?

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In a simple manner, we might say that the the arccos function is basically asking, "What angle has a cosine value of -3.225??"

But the cosine value of an angle can only "range" from -1 to 1, inclusive. So no angle exists with a cosine value of -3.225. That's why you get an "invalid" or "error" message on your calculator. Also, look at the graph of the cosine curve........you'll see that the "crests" are never "greater" than "1," and the "troughs" are never "less" than "-1." 

How many steel plate 120x240 in 30 square meter

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I'm going to assume that the 120 and 240 are in cm....converting these to meters, we have

1.2  and 2.4.  So each steel plate has an area of (1.2m)*(2.4m) = 2.88m²

So....(30m² ) / (2.88m² ) ≈ 10.42 plates......

x/3+y/2=3/2

x/2+y/5=-1/2

?? what is the solution of this ??

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First off, let's simplify things by multipling the first equation on both sides by the common denominator of the fractions..... (6). And let's do the same thing for the second equation for the common denominator of its fractions .........(10).

So we get

2x + 3y  = 9

5x + 2y =  -5

I know there's a solution to this...... (Don't ask me how I  know, I just do!)

If we take the first equation and multiply it on both sides by the "negative" of the coefficient on y in the second equation, (-2), we have

-4x - 6y = -18

Now, take the second equation and multiply it on both sides by the coefficient on y in the first equation, (3). This gives us

15x + 6y = -15

Now, i'm just going to add the first equation to the second...if we do this correctly, we get

11x = -33.........notice how the"y's" just went away!! And solving for, x, we have

x = -3 And putting this back into ANY of the equations will give us "y."

So, using our last equation....

15(-3) + 6y = -15  and

-45 + 6y = -15      add 45 to both sides

6y = 30   so.......y = 5

So x = -3 and y = 5

These are the correct answers.......but don'r just take my word for it........substitute the values back into ANY of the equations and see if they "work'.....(hint: they will)..........

Re: 1+2+3+4+5+6+7+8+9.........+48+49+50+49+48+47.....+5+4+3+2+1

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This looks "tricky," but it's not....we're just adding the first 49 positive integers twice, and then adding 50 to that.

The sum of the first n positive integers is given by (n)(n+1)/2

So, twice that sum is given by 2*[(n)(n+1)/2] = (n)(n+1)

So, twice the sum of the first 49 integers is just (49)(50) and adding 50 to that, we have

(49)(50) + 50   =  50 (49 + 1) = 50(50) = 2500.......

Eugene and Jason like to share stickers. If Eugene gives 70 stickers to Jason, the ratio of Eugene to Jason's stickers will be 1 : 5. If Eugene gives 80 stickers to Jason, their ratio will be 1 : 8. How many stickers does Eugene have?

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Mmmmmm....I always like these kind of brain teasers !!

Let's call the number of tickets that Eugene has (before he gives any of them away) = x

Let's call the number of tickets that Jason has (before he gets any of them) = y

So..when Eugene gives 70 to Jason he has (x-70) and when Jason gets the 70 he has (y + 70)

And the ratio (now) of Eugene's tickets to Jason's tickets is 1/5.

So, this situation is represented by the following proportion:

(x-70)/(y+70) = (1)/(5)

And..when Eugene gives 80 to Jason he has (x-80) and when Jason gets the 80 he has (y + 80)

And the ratio (now) of Eugene's tickets to Jason's tickets is 1/8.

So, this situation is represented by the following proportion:

(x-80)/(y+80) = (1)/(8)

So...taking the first proportion and cross-multiplying we have

5(x-70) = y + 70     .......and simplifying, we get .....  5x - 350 = y + 70  ... and after a little manipulation, we have..... 5x - y = 420

And...taking the second proportion and cross-multiplying we have

8(x-80) = y + 80     ........and simplifying, we get..... 8x - 640 = y + 80  ....and after a little manipulation, we have.....8x - y = 720

Now, I have two equations in two unknowns

8x - y = 720

5x - y = 420

If I take the second equation and subtract it from the frist, I  end up with

3x = 300 ..... and "solving" for  x, we have   x = 100.... (This is the number of stickers that Eugene has)

Now, we could stop at this point, but let's make sure that our answer "works"

Using one of the two equations above, let's "plug in" for x and find out how many tickects that Jason has before he gets any from Eugene.  Using the second equation above, we have

5(100) - y = 420... and simplifying, we have...500 - y = 420 ...and after a little manipulation, we have..........y = 500 - 420 =  80 ....(this is the number of stickers that Jason has before he gets any from Eugene)

Now...one more step.....let's "plug" both values back into both proportions to make sure everything "works"

Using the first proportion, we have

(100-70)/(80+70) = (1)/(5)

30/150 = 1/5....yep...that looks OK

Using the second proportion, we have

(100-80)/(80+80) = (1)/(8)

20/160 = 1/8 ..... and that's OK, too

Whew !!! ...that was a long one....but not really all that difficult....Just remember, "getting" an answer means nothing until we "check" it to make sure that it makes sense!!! .....

Can you assist on how to come up with answer the question is solve 5^(3x-2) = 22 I am coming up with 2.1033 but the answer s/b 2.1333

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Well...let's see

Let's take the log of both sides....this gives

log5^(3x-2) = log(22)   And by a property of logs, we can bring the (3x-2)  "out front"    So we have

(3x-2) log5 = log22  Now, divide both sides by log 5  and we get

(3x-2) = log22/log5  Now, let's add 2 to both sides and divide everything by 3   this gives

x = [ log22/log5 + 2]/3 = 1.3068575534927903

This is the correct answer (verify it for yourself)

2.1333 definitely isn't correct (it results in an answer of about 1190 instead of 22)

2.1033 reults in an answer of about 1029

I tried to "backtrack" to see how you might have made a mistake, but I couldn't find a way to get either of those other two answers.....

what is the result of adding -2.9a + 6.8 and 4.4a - 7,3

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Rom had the correct set up .....(-2.9 + 4.4)a + (6.8 - 7.3), but I think he got side-tracked....the answer is just [1.5a - .5]

What's your question????