geno3141

I think that your answer is better than their's. Having 1500 randomly selected persons is (almost always) more accurate than just 300; but, their answer seems to assume that the reaction of New Jersey residents will reflect the same attitudes as those of the country as a whole; an assumption that, I feel, is not warranted.

By entering -2, you made the y-component to be (y - 5)² which would make the circle tangent to the x-axis --

and the circle will be above the x-axis (center at (6,5)  -- however, you were instructed to translate the circle

downwards -- by making a = 5, you translated the circle upwards.

You will need the y-component to be (y + 5)²  --  which will make the circle tangent to the x-axis -- it will

be tangent below the x-axis.

There are 9 choices for the first (left-hand) digit and 10 choices for the second digit, giving 9 x 10 = 90 

choices for the first two digits.

The last two digits, starting with 00 must be divisible by 4; such as 00, 04, 08, 12, etc.

How many choices are there for the last two digits?

Multiply this answer by 90 to get the final answer for the number of numbers that are divisible by 4.

To find the number that are divisible by 4, subtract that answer from 9999.

By a geometric theorem (involving similar triangles):  CB · CA  =  CD · CL

--->     (7) · (7 + 2x + 3)  =  (9) · (9 + x + 1)

--->          (7) · (10 + 2x)  =  (9) · (10 + x)

--->                  70 + 14x  =  90 + 9x

...

find x; then find AB ...

Hint:     12y³ - 6y²  =  6y²(2y - 1)

If x= 1, y = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 12,13,14,15

If x = 2, y = 1, 2, 3, 4, 5, 6, 7

If x = 3, y = 1, 2, 3, 4, 5

If x = 4, y = 1, 2, 3

If x = 5, y =

...

First term:  a

Second term:  a + d

Third term:  a + 2d

Eleventh term:  a + 10d

Third term is the geometric mean of the first term and the eleventh term:

     a + 2d  =  sqrt( a · (a + 10d) )     --->     (a + 2d)²  =  a(a + 10d)

                                                         a² + 4ad + 4d²  =  a² + 10ad

                                                                 4ad + 4d²  =  10ad

                                                                           4d²  =  6ad

                                                                             2d  =  3a

                                                                               d  =  (3/2)a

Ratio of the second term to the first term: 

     (a + d) / a     --->   ( a + (3/2)a ) / a  

                                       (2a + 3a) / (2a)            (mult num and den by 2)

                                                 5a / 2a

                                                    5/2

My guess:  1, 1, 2, 7, 7, 7, 24   --->   24

I will try to find the white area by adding together the areas of the circles and subtracting out the area of the overlap.

The area of each circle is  pi·6²  =  36pi; so the area of the two circles is  72pi.

Now, for the overlap:

Call the center of the left-hand circle L, the center of the right-hand circle R, the top point of intersection T, and the 

bottom point of intersection B.

Since LT = 6, RT = 6, and LR = 6, angle(LRT) = 60^o and angle(BRT) = 120^o.

To find the segment bounded by arc(TLB) and segment(TB):

area  =  sector(TRB) - triangle(TRB)  =  (120^o/360^o)·36pi - ½·6·6·sin(120^o)

                                                            =           12pi           -    9·sqrt(3)

area of both segments  =  24pi - 18sqrt(3)

Total white area  =  72pi - ( 24pi - 18sqrt(3) )  =  49pi + 18sqrt(3)

Since the triangle is isosceles, the other base angle is also 65^o.

The vertex angle is  180^o - 65^o - 65^o  =  50^o.

x^o + 50^o  =  180^o     --->     x  =  130^o.