All Answers 359165 Answers

Thank you so much!

Thank you so much!

a/b  * b/c    =   a / c  =   sqrt (10)  / sqrt (42)  *   sqrt (135) / sqrt (42)  =

sqrt (2) sqrt (5)                 sqrt (5 ) sqrt (27)

_____________   *         _______________   =

sqrt (2) sqrt (21)              sqrt (2) sqrt (21)

5 sqrt (27)

_____________  =

21 sqrt (2) 

5 sqrt (27) sqrt (2)

_______________   =

21 sqrt (2) sqrt (2)

5 sqrt (54)

_________   = 

   42

5 sqrt (9) sqrt (6)

_____________     =

       42

15 sqrt (6)

________    =

     42

5 sqrt (6)

________

    14

If L3  is parallel to L4, then, by corresponding angles

4x + 10  = 2y       rearrange   as

y  = 2x + 5         (1)

And if L1 is parallel to L2, then, by the property of supplementary angles

5x + 2y - 25 + 2y + x + y    =  180       rearrange as

6x + 5y  =  205     (2)

Sub (1) into (2)

6x + 5 ( 2x + 5)   =  205

6x + 10x + 25   = 205

16x  =  180

x =  180 / 16   =   45/4   = 11.25

And 

y = 2 (11.25) + 5 =   27.5

....or you could just use the sin law

10 / sin 45    =BC / sin 30       solve for BC

10 / sin 45  =  AC / sin 105     solve for AC

https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_14

Draw altitude \(BM\). 

\(BM\) splits \(\triangle ABC\) into 2 triangles: \(30-60-90\) and a \(45 - 45 - 90\) triangle. 

10 is the hypotenuse of the \(30 - 60 - 90\) triangle, meaning \(BM = 5\), and \(AM = 5\sqrt3\)

\(\triangle BCM \) has the angles \(45 - 45- 90\) , so \(CM = 5\), and \(\color{brown}\boxed{BC = 5\sqrt2}\)

We also know \(AC = AM + MC\). Subbing in the known values, we find that \(\color{brown}\boxed{AC = 5 \sqrt3 + 5}\)

We know that \((5x + 2y - 25) + z = 180\). 

Solving for z, we find that \(z = 205 - 5x - 2y\)

Using, alternate interior angles, we know that \(2y + x + y + 205 - 5x - 2y = 180\)

Simplifying, we find that \(25 = 4x - y\)

Because we want \(L_4\) to be parallel to \(L_3\), the angles must be equal, meaning:  \(2y = 4x + 10 \)

We now have a system of equations to solve: \(2y = 4x +10\) and \(25 = 4x - y\)

Solving, we find \(\color{brown}\boxed{x =15, \space y = 35}\)

Ooops....made an error in the last step:edited

Here is another way:

Thx, alan...here's another way using synthetic divsion

-3   [   1    - 3       0         -k                -5       ]

                - 3     18       -54            3k + 162

       _______________________________

          1      -6     18      - k - 54      3k  + 157

So

3k + 157  =  169

3k = 169 - 157

3k = 12

k = 4

Graph:

1534  m/s   +   9 (17) = ___________ m/s

If the two numbers are  a  and  b:

arithmetic mean  =  (a + b) / 2      --->     (a +b) / 2  =  75     --->     a + b  =  150

geometric mean  =  sqrt(a·b)     --->     sqrt(a·b)   =  21     --->     ab  =  441     --->     b  =  441 / a

Combining:  a + 441 / a  =  150    --->     a² + 441  =  150a     --->     a² - 150a + 441  =  0

Using the Quadratic Formula:  a  =  [ 150 +/- sqrt( 20736) ] / 2     --->     147  or  3

a  is one of the two possibilities;  b  is the other.

x = 0 deg, y = 90 deg, max = 2 + a + b.

If two vectors are parallel, their cross product (vector product ) is zero.

So, all three products are zero in which case their sum will be zero.

Here is one way:

Well.... you don't KNOW this one is different as the numbers are missing....BUT the OTHER solutions will show the method....which should be enough to help solve this one !    ~ EP

Uhm, it has not.  It blanked out a number.  It should be, what is the probability that all three resulting pieces are shorter than 3 units?  All of the other ones are not for 3 units.  THis one is different.

This Q has been posted MANY times.... suggest you do a search to get a quick answer  

*6/20 * 2/19 = 12/380 -> 6/190 -> 3/95

4/7 * 3/6 = 12/42 = 2/7

every 5 days  (x) it goes up 5  cases  y     slope =  5/5 = 1

From 12 o'clock the minute hand will be at  90 degrees

   the hour  hand will be at    5 *30 + 30 * 15/60 = 157.25        difference = 67.5 degrees     ( as Guest posted)

p = k/ q²   means that

p q^2 = k    the proportionality constant

19(8^2) = k =1216

p = 1216/ (3²)    = 135.1111