Melody

A real number x is chosen at random such that -2 < x < 5. What is the probability that x^2 < 2? 

if x^2 <2

then

\(-\sqrt2

The distance between these is  \(2\sqrt2\)

The distance between -2 and 5 is 7

so

The prob will be   \(\frac{2\sqrt2}{7}\)

How many 4-letter words with at least one vowel can be constructed from the letters A, B, C, D, and E? (Note that A and E are vowels, any word is valid, not just English language words, and letters may be used more than once.)

Daisy's answer would be correct if the letters could only be used once.

How many 4 letter words can be constructed from ABCDE  that would be 5^4 = 625

How many 4 letter words can be constructed using BCD      that would be 3^4 = 81

So the number of words that use at least one vowel would be  5^4 - 3^4  =  625-81 = 544 

The curve is just made of two identical semi circles.

Together they are the circumference of a circle with diameter 6mm.

Use the formula for the circumference of a circle.

---

The area of the shaded region is just half the area of the big circle.

I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. How many ways are there for me to choose a license plate with these restrictions?

There are 10 digits and 26 letters

you either have

LLD  or    LDD

LLD       26*1*10=260

LDD       26*10*1=260

Total of 520 ways.

It says in the question that x^x^x^x.... =2 

so

x^(x^x^x^x...)  must be  x^2

because x^x^x^x .... =2

so

x^(x^x^x^x..) = x^2

How do you know it is correct?

I assume you are comparing it to an answer that you have, like in an answer sheet.

That does not necessarily mean it is correct, it just means that they are the same.

They might both be wrong    

The answer is fine though   

I know I have not done it your method. I always do them like this.

I think Heureka does it your way. I don't know about other people. 

My way is easy though. ..

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            \(k=2+\sqrt{5}\)