CPhill

Good job, tertre!!!!!

Here's my best attempt.....

I believe that the answer is 1

When two circles are  tangent....a line drawn  through their centers will pass through the point of tangency...so....let the center of the the circle be (x,y)  and the radius = r

The distance for the center of the circle to  (0,5) will just = r

The distance from the center of this circle to the centers of the given circles will be r + 1

We have this system

(x - 4)^2 + y^2 =  (r + 1)^2

(x + 4)^2 + y^2 = (r + 1)^2

x^2 + (y - 5)^2 = r^2

This is a little messy to solve  [ but not impossible]...so I used WolframAlpha to generate the solution

The center of the circle is (x, y) =   (0, 5/3)    and r = 10/3

Here's a graph :

PS....I can show you how to solve the equations if need be...but it's a little lengthy!!!!

We need to find the slope of the line

dy/dx =   dy/dt  /  dx/dt  =  cost / - sin t  =   - cot t

At  t = pi/4, the slope is    - cot (pi/4)  =  - 1

The normal line will have a slope of 1

The point is given by   ( cos pi/4, sin pi/4) =  (  sqrt (2)/2, sqrt (2)/2 )

So....the equation of the tangent lne is

y = 1 (x - sqrt (2) /2 )  +  sqrt(2)/2 

y =  x

Here's the  graph :

Here's my best attempt, Cupcake.......someone else should verify this....

Expected value =

Prob of red * payoff + Prob of green * payoff + Prob of blue * payoff + Prob of yellow * payoff  =

2/12 ( 2)  + 2/12 ( 2) + 2/12 ( 2) +  6/12 (-3)  =   - 1/2

"D"

I believe that....to be a fair game, the payoff for the  yellow should only be losing 2 tokens...

C(50, 6) = 15890700 =  the number of winning "sets" of numbers

Only one will win.....so [ assuming one winner]......the probability of winning is   just 1/15890700

And the probability of losing is  [ 1 - prob of  winning ]

Expected value =

Prize money * probability of winning - Cost to play * probability of  losing =

2,000,000 * (1 / 15890700)   -   2  * [ 1 -   1/15890700]   ≈ -1.87  =   -$1.87

"A store sells cookies at 8 cookies for $2.00. If it changes the price to 10 cookies for $3.00, what is the percent increase in price per cookie?"

2.00 / 8 =  .25 per cookie

3.00 /10  = .30 per cookie

Price increase is

[ 30 - 25 ] / 25  x  100%  =

[ 5 ] / 25    x 100% =

1/5 x 100% =

[100 / 5] %  =

20 %

Two tied 8th notes -  translation and reflection

Two half notes - translation and reflection

The scale factor of the smaller box to the larger   =   ∛ [ 27 /64]  =  3/4

The surface area  of the smaller box =

(surface area of larger box) * (scale factor)^2  =

(160) * (3/4)^2  =

(160) * 9/16  =  

90 in^2

Area = side^2

So

64 = side^2     take the square root of both sides

8 (ft) =  side

8 ft =  8*12  =  96 inches

If each side was 1 ft shorter   =   7 ft on a side

Triangle DBC is  a 3-4-5  right triangle

So  BC = 5

And AB = 20 - 5   =  15

So....since BDis parallel to AE.......we have this relationship

BC/ AB = CD/DE

5/15 = 4/DE

15/5 = DE /4

3 = DE/4   multiply both sides by 4

12 = DE

So 

EC = ED + DC  =  12 + 4  = 16

And by the Pythagorean Theorem,

AE = sqrt [ AC^2 - EC^2 ]  =  sqrt [ 20^2 - 16^2 ] =  sqrt [ 400 - 144] = sqrt [ 256] =  16 ft