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thank you melody it is correct

sin v = 1/√3 and if π/2 < v < π       find tan v

It is in the second quadrant so tan will be negative.

Now I can just 'pretend' it is in the first quadrant for  all the working and then just add the minus sign at the end.

\(sin\; v=\frac{1}{\sqrt3}=\frac{opp}{hyp}\\ \text{Using pythagoras, the adjacent side must be }\sqrt{3-1}=\sqrt2\\ so\\ tan\;v=-\frac{opp}{adj}=-\frac{1}{\sqrt2}\)

Here is a represention of it on a number plane.

Maybe I should also consider it the other way around

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\\ = {1 \pm \sqrt{1+4k} \over 2}\\ \;\;maybe\;\;we \;\;want\\ {1 - \sqrt{1+4k} \over 2} =\left( {1 + \sqrt{1+4k} \over 2}\right)^2\\\)

For this to be true the left hand side has to be positive and we know it is less then \(\frac{1}{2}\)

So that means the RHS before it is squared must be between  \(\pm\frac{1}{\sqrt2}\)

\(2(1-\sqrt{1+4k})=1+1+4k+2\sqrt{1+4k}\\ 1-\sqrt{1+4k}=1+2k+\sqrt{1+4k}\\ -\sqrt{1+4k}=2k+\sqrt{1+4k}\\ -2\sqrt{1+4k}=2k\\ k=-\sqrt{1+4k}\\ \text{If k is a real number then there is not solution here}\)

What if k is complex ??

\(k^2=1+4k\\ k^2-4k-1=0\\ k=\frac{4\pm\sqrt{16+4}}{2}\\ k=2\pm\sqrt{5}\\\)

Well that k is not complex so there is no solution.

So I will stick to my original answer            \(k=2+\sqrt{5}\)

Here is the graph

Mmm

If one root of the equation x^2 - x - k is square that of other find the other root . Find the value of K.

Roots are

    \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\\ = {1 \pm \sqrt{1+4k} \over 2}\\ We \;\;want\\ \left({1 - \sqrt{1+4k} \over 2}\right)^2 = {1 + \sqrt{1+4k} \over 2}\\ \)

The right hand side is clearly bigger than 1 so the left side (before it is squared) must be less than 0

\(1-\sqrt{1+4k}<0\\ \sqrt{1+4k}>1\\ 1+4k>1\\ 4k>0\\ k>0\)

\( \left({1 - \sqrt{1+4k} \over 2}\right)^2 = {1 + \sqrt{1+4k} \over 2}\\ {1+1+4k - 2\sqrt{1+4k} \over 4} = {2 +2 \sqrt{1+4k} \over 4}\\ 2+4k-2\sqrt{1+4k}=2 +2 \sqrt{1+4k}\\ 2k-\sqrt{1+4k}= \sqrt{1+4k}\\ 2k= 2\sqrt{1+4k}\\ k= \sqrt{1+4k}\\ k^2=1+4k\\ k^2-4k-1=0\\ k=\frac{4\pm\sqrt{16+4}}{2}=2\pm\sqrt5\)

\(\\but\;\; k>0\\ so\\ k=2+\sqrt5\)

So there are 5 girls and 3 dresses. 

Each show has 3 girls wearing a dress each.

I'm a bit unclear on the rules.

I guess one or 2 girls can wear the same dress but not all three.

I think it is the same as how many ways can 3 girls be chosen from 5 if order counts.

Order counts becasue the first one gets the first dress, the 2nd gets the 2nd dress and the 3rd gets the 3rd dress.

So that would be      5P3    =  60 

So I think there are 60 shows. Some of the girls and outfits will be the same but never all 3.

Do you have a picture to go with this description?

It doesn't make a lot of sense to me.

\({x}^{{x}^{{x}^{...}}} = 2 \qquad\\ so\\ x^2=2\\ so\\ x=+\sqrt{2}\)

Thanks Daisy,   

Here is just a little more explanation ... I am doing it the long way but it might be easier to understand ... maybe 

A bat and a ball cost one dollar and ten cents in total. The bat costs a dollar more than the ball. How much does the ball cost?

I am going to change the bat to a racket. So I have  R and B

B+100 = R       all in cents,    I can rearrange this to

B-R     =   -100     (1)

I also know that

B+R    =    110     (2)

Now solve simultaneously  -   add

2B+0   =  10

Ball =    5c

We can have the ball equal \(b\). With this, we can set up an equation that says \(2b + 1 = 1.1\). Therefore, \(b = 0.05\). This means that the ball is \(5\) cents. 

- Daisy

The formula for circumference is \(d\pi\).

a) \(d = 18\), so your circumference would be \(18\pi\) inches. Since it is asking to round your answer, we need to multiply \(18\) by \(\pi\), which rounds to \(56.55\) inches. 

b) If \(r = 7\), \(d = 14\). Your circumference would be \(14\pi\) which equals \(43.98\) m.

c) If \(10\) is the diameter, your circumference would be \(10\pi = 31.42\) cm. If \(10\) is the radius, your circumference would be \(20\pi = 62.83\) cm.

d) If \(4.5\) is your diameter, your circumference would be \(4.5\pi = 14.14\) feet. If \(4.5\) is your radius, your circumference would be \(9\pi = 28.27\) feet.

- Daisy

a) The volume formula for a rectangular prism is \(wlh\). To solve this, we can get \((14)(18)(24.5) = 6174\) in³.

b) In order to find how many hinches high the tank is with \(21\) gallons of gasoline, we have to find how much cubic inches of gasoline we have. We can get this from doing \(21*231 = 4851\). Now, we can get an equation to solve for \(x\), the height of the gasoline. We can write the equation as \((18)(24.5)(x)=4851\). By solving the equation, we can get \(x = 11\) inches.

- Daisy

The formula for perimeter/circumference is \(d\pi\), where \(d\) is the diameter. 

a) \(d = 12\), so the circumference is \(12\pi\). Since the question is asking for a semicircle, we divide \(12\pi\) by \(2\) to get \(6\pi\) mm. The question is asking to round your answer, so we can get \(18.85\) mm. 

b) If \(r = 4\), then \(d = 8\). We do the same thing for this problem as we did the last. The circumference for the semicircle would be \(4\pi\) ft. This multiplied by \(\pi\) rounds to \(12.57\) ft.

c) The question does not specify is \(d = 9\) or \(r = 9\). If \(d = 9\), the circumference would be \(4.5\pi\) m. If \(r = 9\), the circumference would be \(9\pi\) m. This will give us \(28.27\) m.

d) This question does not specify whether \(d = 11.5\) or \(r = 11.5\), as well. If \(d = 11.5\), the circumference for the semicircle would be \(5.75\pi\) in. If \(r = 11.5\), the circumference for the semicircle would be \(11.5\pi\) in. So our answer would be \(36.13\) in.

- Daisy

Let's turn \(a\) into \(s\), to not get confused with another \(a\). We can see that \(9x^2 + 24x + s = (ax)^2 + abx + abx + b^2\). This equals \((a^2)x^2 + (2ab)x + b^2\). We have that \(9 = (a^2)\), \(24 = (2ab)\), and \(s = b^2\). By \(9 = (a^2)\), \(a = 3\). If \(a = 3\), we can see that \(b = 4\). Then, we can get that \(s = 16\). So your answer would be \(a = 16\). 

- Daisy

Go online to this site and convert virtually anything you want:

http://www.onlineconversion.com/

There is a cycle of 13 numbers that repeat, counting up from 1 and down to 1. So, we have:

1000 mod 13 = 12th number in the 13-number cycle. Therefore, the 12th number will be a 2 as the 1000th integer.

I agree with the 2 guests.

I believe that the answer is   1/35

I did it the same way as the first guest.   

Ginger you did not take into consideration the number of permutaions for each set of outcomes. You have to think of the 5 dice as all being different from each other.

What is the probability that when we roll 5 fair 6-sided dice, at most 4 of them will show 1?

So there can be 0, 1, 

1- P( all ones)

If all the dice are different then there are  6^5  possible permutations.

There is only 1 way that they can all be 1s

so prob = (1-  1/6^5) = (6^5-1)/6^5 

(6^5-1)/6^5 = 7775/7776

There are ₅C₃ combinations, so we can get \((5\)  \(3)\) \(= 10\) combinations.

- Daisy

thanks

We can see that \(1-2 = -1\), \(3 - 4 = -1\), ... all the way to \(2005-2006 = -1\). \(2007\) has no pair, so we will save that number for later. There are \(1003\) pairs of numbers, which means there are one thousand and three \(-1\)'s. So we have \(-1003 + 2007 = 1004\).

- Daisy

There is \(26\) ways to choose the first/last numbers. Then, there are \(26\) ways to find the second and third letter. Finally, there is \(1 \) way to choose the last number. Multiplying all these numbers, we get \(17576\) combinations of letters. 

- Daisy

Hint: Try to combine the terms!

The probability of an event A occuring given that another event B occured is given by the probability that both events occur divided by the probability that event B occurs.

in this question, A=the event where 2's were rolled on all four dice, and B=the event when the sum of the numbers shown on the dice is 8.

the probability of both events occuring is simply the probability that after rolling, the faces of all dice show 2, which is 1 divided by 6⁴=\(\frac{1}{1296}\)

In the link the guest provided, wolfram alpha gives the final answer of \(\frac{35}{1296}\), so the probability that after rolling the dice, the sum of the numbers on their faces is 8 is 35 divided by 1296

Finally, to find the probability of event A occuring given that event B occured, we have to divide 1/1296 by 35/1296:

\(\frac{\frac{1}{1296}}{\frac{35}{1296}}=\frac{1}{35}\)

so the probability is 1/35

This is a bit tough, but here's my take!

The shortest distance between two circles, is: \(C_1C_2-r_1-r_2\).

We have \(C_1\) as (9,5) as we take out the negatives from (-9,-5), and the radius is 2.5.

And, we have \(C_2\) as (-6,-3) as we add the negatives from (6,3), and the radius is 7.

Now, we use the distance formula! \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) Plugging in the value we get, 

\(\sqrt{(-6-9)^2+(-3-5)^2}=\sqrt{225+64}=\sqrt{289}=17\). After this, we subtract both radii, to attain: 

\(17-2.5-7=17-9.5=\boxed{7.5}\).