CPhill

Answered here :

https://web2.0calc.com/questions/need-help_107

Here might be another way to look at this

Notice that the fractions seem to have the form  n / [ n + 2]    where n is an integer

Suppose that there exists a fraction such that

9/11 < n / [n + 2]  < 11/13 

We can split this into two inequalities

Looking at the inequality on the left....we have

9[n + 2] < 11n     

9n + 18 < 11n

18 < 2n

9 < n    or

n > 9

Looking at the inequality on the right.... we have

13(n) < 11[n + 2]

13n < 11n + 22

2n < 22

n < 11

This implies that

9 < n < 11

So...n = 10 is the only integer that satisfies this

And n + 2 = 12

So....the fraction is 10/12  = 5/6  ......just as tertre found!!!

Good job, tertre   !!!!

\(\frac{3y}{x}+\frac{\left(x+1\right)}{y-1} \)

3y (y - 1) + x(x + 1)

_______________    =

  x (y - 1)

3y^2 - 3y + x^2 + x

_______________

    xy - x

f(x) = 2x^(-1/2) +px

f'(x) = -x^(-3/2) + p

f '(2) = 5       means that

- (2)^(-3/2) + p = 5

-1 / √8 + p = 5

p = [ 5 + 1/√8 ] =

[ 5 + 1 / [ 2√2] ]  =

[ 5 + √2 / 4]  = 

5 + (1/4)√2

3.

Triangle ABE is simlar to Triangle  ECD

AB /AE = EC / ED

AB / (2DE) = EC / DE

AB = 2EC

AB = 2EB

And

AE = 2DE

Therefore.....AD = sqrt ( AE^2 + DE^2) = sqrt [ (2DE)^2 + DE^2) = sqrt(5)DE

So....AD = sqrt(5) (AE/2) = [ sqrt(5)/2] AE

And

AE = sqrt (AB^2 + BE^2) = sqrt [ (2BE)^2 + BE^2] = sqrt(5)BE

BE = AE/sqrt(5)

2BE = 2AE/sqrt(5) = BC

Therfore

AD /BC  =   [ sqrt(5)/2] AE ] / [2AE/sqrt(5)] =  [ sqrt (5) / 2  ] / [ 2 /sqrt(5) ] =

sqrt(5) * sqrt(5) / [ 2 * 2 ] =

5 /4 

Thanks, tertre....that's what I get for watching football and trying to do math at the same time...I'll make the correction....!!!

9/11 = .8181.....

5/6 = .8333....

11/13 = .846....

Mmmmm.....it looks like you could  be correct, tertre.....

Complementary angles sum to 90°

x + 5 + 4x - 15 =  90

5x - 10 = 90      add 10 to both sides

5x = 100           divide by 5

x = 20

Thanks to tertre for spotting my stupid error....!!!

Sorry.....I shouldn't have included "7' in the numerator...we only have a floor of 6 on (2016)^.25

I believe tthe final answer is correct....thanks for noticing my error...

The area is given by

pi * (cone radius) * slant height of cone =

pi * 5 * sqrt (5^2 + 12^2) =

5 * 13 * pi  =

65pi units^2 ≈ 204.2  units^2