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ina 45 - 45 - 90     both legs are the same length = 11 ea

Pythagorean theorom   11^2 + 11^2  = hyp^2       242= hyp^2     hyp = 15.56 in

Thank you so much, CPhill! Wow, amazing! I understand it much better now!

2^2001 * 5^1950

______________  =

     4^27

2^2001 * 5^1950

_____________    =

        (2^2)^27

2^2001 * 5^1950

_____________    =

       2^54

2^1947 * 5^1950 =

2^ 1947 * 5^1947 * 5^3  =

(2*5)^1947  * 5^3  =

10^1947 * 125  =

125 x 10^1947 =

1.25 x 10^1949

So...... 1950 digits

3.  We have that

18 * [  r ]^2  = 8          divide both sides by 18 

 r^2  =  8/18  = 4/9

r^2  = 4/9

r = √[ 4/ 9]   = 2/3

So

The length of the second arc is

18 (2/3)  =  12 ft

4.  ​A little tricky, NSS

We have that

1250 * r^2  = 50     divide both sides by 1250

r^2  =  50/1250  = 1/25

r = √(1/25)  = 1/5

So.....the second term is

1250 (1/5)  =   250

Note that the total amount of money, A, doesn't change.....just the distribution of it does

So....this is a "zero-sum" game....the total amts won + total amts  lost   = 0

7 : 6 : 5

Means that  the first person  started with (7/18)A

The second (6/18)A  =  (1/3)A

The third (5/18)A

In the end   ⇒   6 : 5 : 4

The first person ends up  with (6/15)A  = ( 2/5)A

The second ends  up  with (5/15)A  = (1/3)A

The last ends  up wth (4/15)A

Note that the second person breaks even...he starts with (1/3)A  and ends up with the same amount

The first person must have won the $12  because  (2/5)A > (7/18)A

And the difference between (2/5A - 7/18A)  = 1/90A  =  $12

So.... $12  is 1/90 of the total amount  = $1080

So....the first person must have started with (7/18) * 1080  = $420

And he ends up with (2/5) * 1080  =  $432

So.....he makes $12

{Which means that since the second person breaks even....the third person must have lost the $12 }

Extend  the line CG through to meet AB at Z (say).

The medians of a triangle intersect at a point, so CZ will be the third median, so Z will be the midpoint of AB.

Show that ZG is half the length of AB.

(The easiest way to do this is to draw a line from Z, parallel to AY to meet the line BG. Since Z is the midpoint of AB,

this line will bisect BG. Then consider the small right-angled triangle having ZG as its hypotenuse, use Pythagoras.)

Having done that, use the 1/3, 2/3 property for the median CZ.

Tiggsy

sin x  = opposite / hypotenuse    in a right triangle

remember sin 45 dgrees = sqrt2 / 2

so for the triangle on the LEFT    sin 45 = sqrt(2) /2  = a/(7 sqrt(2) )  solve for a

                                                                  (7 * sqrt2) * sqrt(2)/2 = a

                                                                   14/2 = a = 7

1)  input would be   hours worked     output would be  $ earned

2)  input would be time of day       output   lenght of shadow

3) input   gallons of gas      out put   distance driven

(Often, input is graphed as 'x'    output is  'y'   ) 

1.  No...in a geometric series.the ratio between the successive term to the previous term must be constant ....note that

-4 / 2   =   -2

But

-16 / - 4  =  4    so....not geometric

2. Fifth term  

The common ratio  is  36/120  =  3/10

So....the fifth term is

First Term * (Common Ratio)^n-1   =

120 (3/10)^5-1  =

120 (3/10)⁴   = 

 0.972

In vertex form:

y = x^2 +8x  +16     -16 +18     ('completing the square)

y = (x+4)^2 +2            Vertex  h, k   = -4,2    Upward opening parabola (coefficient of (x+4)^2 is positive: +1  )

Re-arranging

y-2 = 4p (x+4)^2          4p = 1   p= 1/4   distance from vertex to directrix and focus

Directrix =   y= 2- 1/4 = 1.75

Focus = y= 2+1/4 = 2 1/4       x = -4        so   -4, 2 1/4

Graph:

I think the above answer is correct, because it just happens that:

The LCM of {7, 6, 5, 4} =420 and the ratio being 7:6:5 or:

$420:$360:$300 =$1,080.

Since $1,080 / [7+6+5] =$60 average per man when they started gambling, and:

Since $1080 /  [6+5+4] =$72 average per man if they hadn't lost any money, or:

$432:$360:$288, but one of them ended up with:

$432 - $420 =$12 for the first man, for the ratio of:

$12:$10:$8 =$30 - total wininings for the 3 men, or is it:

$300 - $288 =$12 - Left by the last man, in which case the ratio will be:

$18:$15:$12 =$45 - total winnings for 3 men ????. This last one seems to make more sense, but can't be right because that would mean that he started with $300, which is NOT one of the answers!!!.

LOL!!!!

you answered the second by answering the first one :D

Can't answer the second one....but....for the first

The vertex  is (0,0)

This parabola opens to the left

The general form  for this type of parabola with the vertex is at the origin  is

-4px  = y^2         where p is the distance between the focus and the vertex

So....we have

x =  -4y^2        divide both sides by -4

(-1/4)x  = y^2

So.......this implies that  (-1/4) = -4p  ⇒  p = 1/16

So.....the focus in this situation is given by

(0 - p , 0)  =  (0 - 1/16 , 0 )  =  (-1/16, 0 )

And the directrix  is give by:    x = 1/16

Here's a graph :  https://www.desmos.com/calculator/z66mdaucdo 

Note that when a = 2

P(x  ≤ 2)   =  the probability of this area  =  1/2  of the area of the figure  =  0.5

So....a  = 2

1.  Sum  =   ( First term + Last term) * (number of terms) / 2  =

(44 + 26) * (7/2)  =

(70) / 2 * 7 =

35 * 7  =

245

2. Similarly

(7.6 +   -1.5) * 8 /2  =

(6.1) * 4  =

24.4

Area of trapezoid  =

height / 2  * (sum of base lengths) =

(2/10) / 2 * ( 8 + 2)  =

(2/20) * 10   =

20/20  =  

1 square unit   = Total Area

P ( x  ≤ 3)  

We have a trangle with  a height of (2/10)  and base = 3

Area  = (1/2)(3)(2/10)  = 3 / 10

So....P ( x  ≤ 3)    =   ( Area of triangle) / (Total Area )  =

(3/10) / 1  =   3/10 =   0.3

First one....we have a trapezoid.....

The area of a trapezoid  is

height/ 2 * ( sum of base lengths)

So we have......

.25 / 2  *  ( 6 + 2)  =  8/8   =    1  unit^2

And we are looking for areas =  .25 of this  =1/4 of this  =  .25   units^2

P( x ≥ 5)  =   area of a triangle with base = 1  and height  .25/2

Area  (1/2)base * height  =  (1/2) * 1 (.25 /2)  = .25/ 4  =  .0625  No

P ( 3 ≤ x ≤ 4)  = Area of rectangle = .25 * 1  = .25    Yes     (1)

P (  x ≤2 )  = Triangle with base of 2 and height of .25  =

(1/2)(2) * .25  =   .25  Yes

P ( 2 ≤ x ≤ 3)....same area as (1)   =  .25

So....only  the first answer is incorrect 

{9, 8, 7} | {9, 8, 6} | {9, 8, 5} | {9, 8, 4} | {9, 8, 3} | {9, 8, 2} | {9, 8, 1} | {9, 8, 0} | {9, 7, 6} | {9, 7, 5} | {9, 7, 4} | {9, 7, 3} | {9, 7, 2} | {9, 7, 1} | {9, 7, 0} | {9, 6, 5} | {9, 6, 4} | {9, 6, 3} | {9, 6, 2} | {9, 6, 1} | {9, 6, 0} | {9, 5, 4} | {9, 5, 3} | {9, 5, 2} | {9, 5, 1} | {9, 5, 0} | {9, 4, 3} | {9, 4, 2} | {9, 4, 1} | {9, 4, 0} | {9, 3, 2} | {9, 3, 1} | {9, 3, 0} | {9, 2, 1} | {9, 2, 0} | {9, 1, 0} | {8, 7, 6} | {8, 7, 5} | {8, 7, 4} | {8, 7, 3} | {8, 7, 2} | {8, 7, 1} | {8, 7, 0} | {8, 6, 5} | {8, 6, 4} | {8, 6, 3} | {8, 6, 2} | {8, 6, 1} | {8, 6, 0} | {8, 5, 4} | {8, 5, 3} | {8, 5, 2} | {8, 5, 1} | {8, 5, 0} | {8, 4, 3} | {8, 4, 2} | {8, 4, 1} | {8, 4, 0} | {8, 3, 2} | {8, 3, 1} | {8, 3, 0} | {8, 2, 1} | {8, 2, 0} | {8, 1, 0} | {7, 6, 5} | {7, 6, 4} | {7, 6, 3} | {7, 6, 2} | {7, 6, 1} | {7, 6, 0} | {7, 5, 4} | {7, 5, 3} | {7, 5, 2} | {7, 5, 1} | {7, 5, 0} | {7, 4, 3} | {7, 4, 2} | {7, 4, 1} | {7, 4, 0} | {7, 3, 2} | {7, 3, 1} | {7, 3, 0} | {7, 2, 1} | {7, 2, 0} | {7, 1, 0} | {6, 5, 4} | {6, 5, 3} | {6, 5, 2} | {6, 5, 1} | {6, 5, 0} | {6, 4, 3} | {6, 4, 2} | {6, 4, 1} | {6, 4, 0} | {6, 3, 2} | {6, 3, 1} | {6, 3, 0} | {6, 2, 1} | {6, 2, 0} | {6, 1, 0} | {5, 4, 3} | {5, 4, 2} | {5, 4, 1} | {5, 4, 0} | {5, 3, 2} | {5, 3, 1} | {5, 3, 0} | {5, 2, 1} | {5, 2, 0} | {5, 1, 0} | {4, 3, 2} | {4, 3, 1} | {4, 3, 0} l {4, 2, 1} | {4, 2, 0} | {4, 1, 0} | {3, 2, 1} | {3, 2, 0} | {3, 1, 0} | {2, 1, 0} {120 Numbers}

3.

Note NSS  that the distances  are  26, 44, 62, 80.....  each successive term is 18 more than the succeeding one

The 7th term  is

26 +  18 ( 7 - 1)   =   26 + 18 * 6   =  134

For an arithmetic sequence.....the sum is given by :

 ( First term + Last term ) *  number of terms / 2  =

(26 + 134) * 7 /2    =   160 * 7 / 2   =  160 / 2 * 7   =  80 * 7  =  560 ft

EP's  answer to the second one is good

And for his third answer     5 + 6 + 7 + 8  =

[ 5 + 8] * 4 / 2   =   13 * 2    = 26  

First one

P ( 2.5   ≤ x   ≤ 4 )  =

Width of the interval of interest

_________________________     =

    Total width of interval

(4 - 2.5)                   1.5

______     =        ________    =   .375

  4 - 0                       4

Second one

P( < 180)  =

180 - 177                  3

_________  =         ___    =   .60

182  - 177                 5

We have

4px  =  y^2         p  = distance between the origin and  the directrix

4(5/2)x = y^2             

10x = y^2

x  = (1/10)y^2

a₁ = 6; a_n = 1/4_an−1

The explicit rule is

aⁿ  =  6(1/4)^n-1

1)   You can just add all of the numbers to get the total....use a calculator

     or use      sum =  n ( a1+an) / 2   =   7 (26+44)/2  =  ??         (n= number of terms  a1=first term  an = nth term)

2.)   add them all up with calculator....or use

        Sum = n (a1+an)/2   =  8 (7.6 + (-1.5) )/2 = ??