geno3141

In a geometric progression:     n₁  =  7     and     n₆  =  7/32

Since  n₆  =  n₁ · r⁵      --->     7/32  =  7 · r⁵     

                                     --->     1/32  =  r⁵     (divide by 7)

                                     --->       1/2  =  r       (take the fifth root)

a  =  7 · (1/2)

b  =  7 · (1/2)²  

The shortest distance from a point to a line is the perpendicular distance from that point to that line.

Step 1:  Find the slope of the line  -6x + 5y - 1  =  0.  Call this slope m₁.

Step 2:  Find the negative reciprocal of that slope; call this slope m₂.

                 This is the slope of the line that is perpendicular to the original line.

Step 3:  Use the point-slope form ( m₂ and the point (-3,1) ) to find the equation of the line through

                 the point (-3,1) that is perpendicular to the original line.

Step 4:  Using the original equation and the equation that you found in step 3, find the point of 

                 intersection of these two lines. This is the answer.

1)  EF || GH   --->   arc(EH) = arc(FG)   --->   x² - 2x  =  56 - 3x

                                                                  x² + x - 56  =  0

                                                               (x + 8)(x - 7)  =  0

                                                                                 x  =  7                      (can't be negative)

    I don't know how to use this value for x to find arc(EPF) because I don't know where P is.

2)  Because this is a regular dodecagon, each side is congruent to every other side.

     Since you haven't had any trig, try this:

     Within the regular dodecagon,

     --  Draw the regular hexagon  P₁P₃P₅P₇P₉P₁₁  (with the center of the circle = O)

          Since it is a regular hexagon, angle(P₁OP₃)  =  60^o      

          OP₁  =  OP₃ =  P₁P₃  =  1

     --  Draw the diameter of the circle that goes through P₂.

          This bisects angle(P₁OP₃) and also bisects P₁P₃  --  call this point X.

    --  P₁X · XP₃  =  P₂X · XP₈        (P₈ is the other end of the diameter from P₂)

         P₁X  =  XP₃  =  1/2

         Let P₂X  =  x   --->   XP₈  =  2 - x

         P₁X · XP₃  =  P₂X · XP₈   --->   (1/2)(1/2)  =  (x)(2 - x)

                                                                     1/4  =  2x - x²

                                                        x² - 2x - 1/4  =  0

          By the quadratic formula:  x  =  ( 2 - sqrt(3) ) / 2

       Now, look at triangle P₁XP₂  --  it is a right triangle with P₁P₂ the hypotenuse

       Using the Pythagorean Theorem:  (P₁P₂)²  =  (1/2)² + ( 2 - sqrt(3) )²   

                    --->   P₁P₂  =  2 - sqrt(3)

To find your answer, you'll need to square this and multiply by 12.

40% of 25000  =  0.40 x 25000  =  10000

30% of 10000  =  0.30 x 10000  =  3000

20% of 3000  =  0.20 x 3000  =  600

10% of 600  =  0.10 x 600  =  60

More simply, just do this:  0.10 x 0.20 x 0.30 x 0.40 x 25000  =  60

Generally speaking, you can't turn an  x - 1  into an  x + 1.

You can turn a  1 - x  into an  x - 1  by factoring out a negative one:  1 - x  =  -1(x - 1)

  x = 2.5    --->    2

2x = 5.0    --->    5

3x = 7.5    --->    7

Let  3x  be the number of girls 

and 2x  be the number of boys.

Seven more girls join the class:  the number of girls is now  3x + 7

Two boys drop the class:            the number of boys is now  2x - 2.

The ratio of girls to boys is now 5:2     --->     (3x + 7) : (2x - 2)  =  5 : 2

         (3x + 7) / (2x - 2)  =  5 / 2

                  2 · (3x + 7)  =  5 · (2x - 2)

                         6x + 14  =  10x - 10

                                 24  =  4x

                                   6  =  x

Current number of girls:  3x + 7   --->   3(6) + 7  =  25

Current number of boys:  2x - 2   --->   2(6) - 2  =  10

Starting with:  e^i·x  =  cos(x) + i·sin(x)

If  x  =  pi/2   --->   e^i·(pi/2)  =  cos( pi/2 ) + i·sin( pi/2 )

                    --->   e^i·(pi/2)  =  0 + i·1  =  i     --->     i  =   e^i·(pi/2) 

So:  iⁱ  =  ( e^i·(pi/2) )ⁱ  =  e^{i^2 · pi/2}  =  e^-1·pi/2  =  e^{- pi/2}  

Let      log_x3  =  a     --->        x^a  =  3                           (equation 1)

Also:  log₈₁x  =  a     --->     81^a  =  x                            (equation 2)

Substitue equation 2 into equation 1:  (81^a)^a  =  3     --->     81^{a^2}  =  3

Since  81  =  3⁴     --->     (3⁴)^{a^2}  =  3     --->     3^{4a^2}  =  3¹  

                                                                --->        4a²  =  1

                                                                --->          a²  =  ¼

                                                                --->           a  =  ½

Using equation 1:  x^a  =  3     --->     x^½  =  3     --->     x  =  9

If Axex makes 60% of his shots, then he misses 40% of his shots.

Assuming that each of his shots is independent from each of his other shots, then it is correct to

multiply the probabilities together:  0.40 x 0.40  =  0.16