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Thanks, I bumped into the same promblem and this helped alot.

A bag contains yellow, green, blue, and white marbles.  The ratio of yellow, green, blue, and white marbles is 1:2:3:4.  If the bag contains 180 marbles, then how many blue marbles are there?  

Consider:  "The ratio of yellow, green, blue, and white marbles is 1:2:3:4."  

Call the number of yellow marbles X.  

So the number of green marbles is 2X. 

And the number of blue marbles is 3X.  

And the number of white marbles is 4X.  

Add up all the marbles and you have     10X  =  180  

Therefore                                                   X  =  18 yellow marbles  

                                                                 2X  =  36 green marbles  

                                                                 3X  =  54 blue marbles  

                                                                 4X  =  72 white marbles

Hope this helps! 

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Thanks, I think I might be able to solve the next promblem.

We can find k by substituting the known values of x and y into the equation.

3\sqrt{2} = k(81)^(1/4)

3\sqrt{2} = k(3)

k = 3\sqrt{2}/3 = \sqrt{2}

Now we can find y at x=4 by substituting this value of k into the equation.

y = \sqrt{2}(4)^(1/4) = \sqrt{2}(2) = 2\sqrt{2}

Therefore, the value of y at x=4 is 2\sqrt{2}.

Could you format the question a little better please?

The probability of drawing an even number is 3/7. There are 3 even cards (2, 4, and 6) out of 7 total cards.

The probability of drawing a multiple of 3 is 2/7. There are 2 multiples of 3 (3 and 6) out of 7 total cards.

The probability of both of these events happening is 3/7 * 2/7 = 6/49.

Therefore, the probability that Professor Grok draws a card with an even number, and then a card that is a multiple of 3, is 6/49.

Here is the solution in more detail:

The probability of event A happening is the number of ways event A can happen divided by the total number of possible outcomes. In this case, event A is drawing a card with an even number. There are 3 even cards in the deck, so the probability of drawing an even card is 3/7.

The probability of event B happening is the number of ways event B can happen divided by the total number of possible outcomes after event A has already happened. In this case, event B is drawing a card that is a multiple of 3. There are 2 multiples of 3 in the deck, but one of them has already been drawn, so the probability of drawing a card that is a multiple of 3 after already drawing an even card is 2/6 = 2/7.

The probability of both events happening is the product of the probabilities of each event happening. In this case, the probability of drawing a card with an even number and then a card that is a multiple of 3 is 3/7 * 2/7 = 6/49.

3 is immediately seen obviously incorrect because  

the sqrt(12) by itself is already more than 3, before  

you start adding all those other things to it.    

(e^2+1+2e/4) *   e^-1  

This needs some more parentheses to clarify exactly what is wanted.  

The problem I'm having specifically is the 2e/4 term.  

Why wouldn't that have just been called e/2 instead.  

The only reason I can think of for writing 2e/4 instead of simply e/2   

is that the denominator 4 applies to everything that comes before.  

Thusly                                         e² + 1 + 2e         e^–1  

                                                  —–————  •  ———  

                                                           4                  1  

That can be written as                 (e + 1)²   

                                                   –––––––  

                                                        4e  

I don't know if this is correct ... "correct" meaning what was  

actually wanted, so I'm not going to paint it red like I usually do.  

_.

2000/40=6000/x

240000=2000x

120=x

120 minutes!

The answer is [200,470).

That is incorrect.

Thanks,
Guest

=12.5916

The length of QR is 64.

It's either "a" or "not b". You can try it right now

Well, the idea is the same:

\(x=\sqrt{12-\sqrt{12-\sqrt{12-...}}}\\ \iff x^2=12-x \\ \iff x^2+x-12=0 \\ \iff (x+4)(x-3)=0 \\ \iff x=3\)

We are given that square ABCD, P is on BC such that BP = 4 and PC = 1, and Q is on CD such that DQ= 4 and QC = 1. This means that triangle BPQ and triangle CQP are both right triangles.

We are asked to find sin angle PAQ. We can do this by using the following steps:

Find the length of side PQ.

Use the Pythagorean Theorem to find the length of side PA.

Use the ratio of opposite to hypotenuse to find sin angle PAQ.

Step 1:

The length of side PQ can be found by using the Pythagorean Theorem on triangle BPQ.

PQ^2 = BP^2 + PC^2 PQ^2 = 4^2 + 1^2 PQ^2 = 17 PQ = sqrt(17)

Step 2:

The length of side PA can be found by using the Pythagorean Theorem on triangle CQP.

PA^2 = CQ^2 + QP^2 PA^2 = 4^2 + sqrt(17)^2 PA^2 = 33 PA = sqrt(33)

Step 3:

The sine of angle PAQ can be found by using the ratio of opposite to hypotenuse.

sin angle PAQ = PQ / PA sin angle PAQ = sqrt(17) / sqrt(33) sin angle PAQ = sqrt(17/33)

Therefore, the sin angle PAQ is sqrt(17/33).

Find the ordered triple (p,q,r) that satisfies the following system:  

p - 2q = 3  

q - 2r = -2 + q  

p + r = 9 + p  

Per the second line       –2r = –2  therefore  r = 1  

Per the third line,                                          r = 9  

You can't have r equal two different amounts,  

so the system as stated can not be solved.   

_.

The answer is "a or not b"

I greatly appreciate your effort to help, however, I belive I found an error in your reasoning.

When you say "The side length of the cube is the distance from O to the opposite vertex." that does not really make sense since the diagonal of a cube is x\(\sqrt{3} \) 

Also I think the whole solutions feels rather strange, from what I tried using similar triangles gets me farther

When the problem is "Minus Sqrt(12)", the answer is 3 and not 2.

Will and Grace are canoeing on a lake.  Will rows at 50 meters per minute and Grace rows at 20 meters per minute. Will starts rowing at 2 p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at 2 p.m. If they always row directly towards each other, and the lake is 3800 meters across from the west side of the lake to the east side, at what time will the two meet?  

Will's rate is 50 m/min.  

Grace's rate is 20 m/min.  

They're rowing toward each other,  

so their rate of closure is 70 m/min.  

The lake is 3800 meters across.  

The time it will take them to meet is given by  

                                                                                    Distance  

                                                                            T  =  –––––––  

                                                                                       Rate  

                                                                                    3800 m  

                                                                            T  =  –––––––  

                                                                                    70 m/min  

The time it takes them to meet                             T  =  54.29 min  

Note that (0.29 min • 60 sec/min)  =  17 seconds

The time on the clock is                                             2:54:17 pm 

_.

A bag contains yellow, green, blue, and white marbles.  The ratio of yellow, green, blue, and white marbles is 1:2:3:4.  If the bag contains 180 marbles, then how many blue marbles are there?  

Consider:  "The ratio of yellow, green, blue, and white marbles is 1:2:3:4."  

Call the number of yellow marbles X.  

So the number of green marbles is 2X. 

And the number of blue marbles is 3X.  

And the number of white marbles is 4X.  

Add up all the marbles and you have     10X  =  180  

Therefore                                                   X  =  18 yellow marbles  

                                                                 2X  =  36 green marbles  

                                                                 3X  =  54 blue marbles  

                                                                 4X  =  72 white marbles  

_.