geno3141

h(x)  =  2x(x - 5)³(x + 1)(x - 2)²

Degree of 2x:           1

                (x - 5)³:     3

                (x + 1):      1

                (x - 2)²:     2

Adding together these degrees:  final degree = 7

The graph bounces at -6, so it has a factor of (x + 6)².

The graph passes through -3, so it has a factor of (x + 3)

The graph bounces through 0, so it has a factor of x².

The graph has this type of equation:  f(x)  =  a(x + 6)²(x + 3)(x²)

To see if a is positive or negative, check the end behavior:

-- if you put in a very large positive number, (x + 6)²(x + 3)(x²) will be positive (which means that the graph rises on the right-end

-- however, it drops on the right-end, so a must be negative.

Answer:  f(x)  =  -(x + 6)²(x + 3)(x²)

Now, if you have to make certain that it hits all the other points, we would have to do a lot more work ...

y  >  0.5x - 0.5

multiply by 2:

2y > x - 1

-x + 2y + 1 > 0

x - 2y - 1 < 0

Sum of an infinite series:  Sum  =  a / (1 - r)

First series:  common ratio =  4 / 12  =  1/3      --->     Sum  =  12 / (1 - 1/3)  =  12 / (2/3  =  18

Second series:  common ratio  =  (4 + n) / 12     --->     Sum  =  12 / ( 1 - (4 + n)/12 )

--->     12 / ( 1 - (4 + n)/12 )  =  4 · 18

--->     12 / ( 1 - (4 + n)/12 )  =  72

multiplying the numerator and denominator on the left side by 12/12

--->     144 / ( 12 - (4 + n) )  =  72

--->     144 / ( 8 - n )  =  72

--->     144  =  72(8 - n)

--->     144  =  576 - 8n

--->    -432  =  -8n

--->         n  =  6

You start with 150L of a mixture.

You drain out 40L, leaving you with 110L.

This is in the ratio (water to alcohol) of 2:3.

Therefore 2/5 of the 110L is water:  2/5 x 110  =  44L of water.

and 3/5 of the 110L is alcohol:  3/5 x 110  =  66L of alcohol.

To this you add 24L of alcohol and 12L of water.

You now have a mixture of 44 + 12 = 66L of water and 66 + 24 = 90L of alcohol.

water : alcohol  =  66 : 90  =  11 : 15

Since angle(R) is exterior to the circle it will be the difference of the two intercepted arcs divided by 2.

angle(R)  =  ( arc(SQ) - arc(TU) ) / 2  =  ( 103 - 43 ) / 2  =  60 / 2  =  30.

The average (mean) of four numbers is their sum divided by 4:  144 / 4  =  36

Since TA and TB are tangents from the same point, TA = TB = x.

Since BU and UC are tangents from the same point, BU = UC = x.

Since CR and RD are tangents from the same point, CR = RD = 3.

Since SD and SA are tangents from the same point, SD = SA = 3.

TA + TB + BU + UC + CR + RD + SD + SA  =  18

   x + x   +    x + x     +    3 + 3     +   3 + 3     =  18

Can you finish this?

Parallel lines:     y  =  (3a + 2)x - 2

                         2y  =  (a - 4)x + 2     --->     y  =  [ (a - 4)/2 ]x + 1

Since the lines are parallel, the slopes are equal:

          3a + 2  =  (a - 4)/2

          6a + 4  =  a - 4

                5a  =  - 8

                  a  =  -8/5

In triangle(ABC), AB = 13, BC = 14, and AC = 15.   

M is the midpoint of BC.

Find AM.

I'm going to use the Law of Cosines twice  

-- first, with triangle(ABC), to find angle(C)

-- and then, with triangle(ABM), to find AM.

Triangle(ABC):  AC2  =  AB2 + BC2 - 2·AB·BC·cos(B)

--->                     152  =  132 + 142 - 2·13·14·cos(B)

--->                     225  =  169 + 196 - 364·cos(B)

--->                    -140  =  -364·cos(B)

--->                 cos(B)  =  0.3846  

--->                        B  =  67.38o  

Triangle(ABM)  AM2  =  AB2 + BM2 - 2·AB·BC·cos(B)

--->                   AM2  =  132 + 72 - 2·13·7·cos(67.38o)

--->                   AM2  =  148

--->                   AM  =  sqrt(148)