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A plane flies from Alphaville to Betaville and then back to Alphaville. When there is no wind, the round trip takes 4 hours and 48 minutes, but when there is a wind blowing from Alphaville to Betaville at 100 miles per hour, the trip takes 5 hours. How many miles is the distance from Alphaville to Betaville?

4hrs, 48 min    =  4.8 hrs.

Let D be the distance  between the towns, and  let R be the  normal rate of the plane

So we have this equation [ with no wind ]

2D / R  =   4.8

Solving for  D  we have that

2D = 4.8R

D =  2.4R    (1)

And when the wind is blowing, we have this equation

D / (R + 100)  + D / ( R - 100)  =  5     simplify

[ D(R - 100) + D(R + 100) ] / [ (R + 100)(R - 100) ]  = 5

[ 2DR]  = 5 [ (R + 100) (R - 100) ]       sub  (1)  for D 

2 (2.4R)R  =  5 (R^2 - 10000)

4.8R^2  = 5R^2  - 50000

5R^2 - 4.8R^2 - 50000  = 0

.2R^2  - 50000  = 0

.2R^2   =  50000

R^2  = 50000/ .2   

R^2 =  50000 * 10/ 2

R^2  = 250,000     take the square root

R = 500 

And  using (1 )the distance between the towns  is  (2.4)(500) = 1200 miles

Nice, Alan.....

It will leak  at  3 * 60 minutes   = 180  oz per hour

So

180  oz / hr   *    1 gal / 128 oz       =    (cancel the units)

180 / 128 gal / hr   =

45 / 32  gal / hr  =

1.40625 gal / hr

Great! Thanks guys!

If we increase 10  by 100%....we effectively double 10  = 20

If we increase 10 by 150%....we add  15 to 10   = 25

So.....150%

Ok thanks. I found my mistake. I was doing it too quickly...

It is NEITHER !!!.

Write this rule down for the future:

25/10 =2.5 - 1  x  100 =1.5  x  100 = 150% increase.

Very nice, heureka!!!!

Here is another way:
a) 4sin(x) = cos(x)                     divide both sides by cos(x)
4[sin(x) / cos(x)] =1
4 tan(x) = 1                                divide both sides by 4
tan(x) = 1 / 4
x =arctan(1/4) =~14.04 deg.     Then proceed as CPhill did.

Oh, thank you, CPhill! My small error, oops!

Almost, terte

9n + 4   = 0   ⇒   n  = -4/9    and this is what we need to divide by

-4/9   [  9    49    - 7   - 102    - 41  ]

                   -4    -20      12       40

         ________________________

           9    45     -27     -90      - 1

Your remainder of  -1   is correct

The apparent residual plynomial is    9n^3  + 45n^2  - 27n  - 90

Note....to find the correct residual polynomial,  divide  the  apparent  residual polynomial by the  "9"  in the linear divisor of  9x - 4

So...we actually have

n^3  + 5n^2  - 3n  - 10  R ( -1/ [ 9n  - 4 ] )

To see why this is true...note the polynomial division  ;

                  n^3

(9n - 4)  [ 9n^4.........            ]

               -(9n^4)

               ___________

etc. ........

Note that the first term is  n^3 .....not 9n^3...

the word is modulus

\(|Z| = \left | \dfrac 1 Z \right | = | 1 - Z |\)

first off you should know that

 \(\left | \dfrac 1 Z \right | = \dfrac {1}{| Z |} \\ \\ \text{so } |Z| = \left|\dfrac 1 Z \right| \Rightarrow |Z|=1\)

\(|1-Z|=1 \\ \\ \sqrt{(1-Z)(1-Z^*)} = 1 \\ \\ (1-Z)(1-Z^*)=1 \\ \\ 1 -Z -Z^* + Z Z^* = 1\)

\(\text{but }Z Z^* = |Z|^2 = 1 \text{ so} \\ 1-Z-Z^* + 1 = 1 \\ Z+Z^* = 1 \\ 2 Re(Z) = 1 \\ Re(Z) = \dfrac 1 2\)

\(1 = 1^2 = |Z|^2 = Re(Z)^2 + Im(Z)^2 = \left(\dfrac 1 2\right)^2 + Im(Z)^2 \\ \\ Im(Z)^2 = \dfrac 3 4 \\ \\ Im(Z) = \pm \dfrac{\sqrt{3}}{2} \\ \\ Z = \dfrac 1 2 (1 \pm i\sqrt{3})\)

Well, I'll solve one of them.

1. \(6x+4x-6=24+9x, 10x-6=24+9x, x=30\), so one solution.

Remember infinite solutions, 0=0, etc.

Tack  !!!

Thanks, CPhill!

Divide 9n^4+49n^3-7n^2-102n-41 by -4/9, 9  49  -7 -102 -41 divided by -4/9,

Solving synthetic, 9x^3+45x^2-27x-90-1/x-4/9?

Is it wrong?

Glad to help  !!!

Thank you so much! I will be able to do a lot better now :D

Nice, tertre  !!!

-5/2(3x+4)<6-3x        distribute the   -5/2  over the terms in the parentheses

(-15/2)x - 20/2  < 6 - 3x

(-15/2)x  - 10  <  6 - 3x         add 3x , 10 to both sides

(-15/2)x + 3x  <  6 + 10 

(-15/2)x  + (6/2)x   < 16

(-9/2)x <  16        multiply both sides by  -2/9....change the direction of the inequality sign

x > 16 (-2/9)

x > -32/ 9

Correct,RP!!!...good job  !!!

[Just a note...remember to write the remainder as   -1 /[9n + 4]....what you have is  -1/9n + 4...which translates to :

(-1/9) n   +  4 .... ]

Otherwise....very nice!!!

Wow! Great Solution, CPhill! 

Let x represent the cost of one of Tanya's items, and y represent the cost of one of Tony's items. So, we have y+1.25=x.

So, 4(y+1.25)=5(y), 4y+5=5y, y=5. And, x=5+1.25=$6.25. Tanya-6.25*4=25, Tony-5*5=$25.

Different roads....same answer....good job  !!!

[You divided the polynomial by 8 right away.....and realized that the  "7"  in the denominator of the "remainder fraction" represented the true remainder....if you like that method better, stick with it...I just learned it a different way....at least you understand these, now....that's the important thing  !!! ]