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Nice, PM....!!!!

BTW.....we don't need the triangle, at all....

The mean is 21.75

Std deviation is 7.33

One std deviation from the mean is between   14.42   and  29.08

14 values fall within this...so  14/20 =  70%

Two std deviation lie between  7.09 and 14.42  and from 29.08 and 36.41

5 values fall here  = 5/20  = 1/4  = 25%

1 value falls 3 sd deviations from the mean  = 1/20  = 5%  of data

This data is close to  being normally distributed

70% falls within 1 std deviationn......68% is expected

95% falls wihin 2 std deviations......exactly what we expect

And 100% fall within 3 std deviations.....99% is expected

ok, I got it. It is 48/%

Thanks

\(\boxed{48\%}\)

The side length of the yellow square is 4, so the area of the yellow square is 16 (units^2).

The total area of the green is the [Area of Green Square] - [Area of Yellow Square]. The side length of the green square is 8, so with the yellow square inside, the area is 64 (units^2). 64 - 16 = 48 (units^2). 

The probability of the dart hitting the green area is (Green Area) / (Total Area), which is \(\dfrac{48}{64} = \boxed{0.75}\). The answer is C) 0.75. 

- PM

Here it is, PM: \(\boxed{48\%}\)

We can use the variables μ (mu) to represent the length of the horn, and σ (sigma) for the standard deviation.

The average horn spread:  μ = 60 inches.

The standard deviation:  σ = 4.5 inches.

According to the normal distribution, the range for the middle 95% of the values should be: [ μ  - 2σ,  μ + 2σ ]

Replacing the given values, we have [ 60 inches  - 2(4.5 inches), 60 inches + 2(4.5 inches) ]. 

Distributing out the values, we have [ 60 inches  - 9 inches, 60 inches + 9 inches ].

Our solution is [\(\boxed{ 51 \text{inches,} 69 \text{inches} }\)]. (with the [] brackets)

- PM

The answer is 48%. The Latex for the percent sign does not work.

John's probability to be on time: 100% - 20% = 80% or 0.8.

Ted's probability to be on time: 100% - 40% = 60% or 0.6.

To find the probability that both will be on time, we multiply the two probabilities. \(0.8\cdot0.6 = 0.48\). The answer is \(\boxed{48}\)%.

- PM

The units digit of a number that is 4 more than a multiple of 5 must be 4 or 9. We can just try a few numbers starting from 4. We can try 4, 9, 14, 19, 24, etc. 

4 doesn't work because 4 = multiple of 3 +1.

9 doesn't work because 9 is multiple of 3

14 does work because 14 = multiple of 3 + 2. 14 = 12 + 2. 

The smallest value of X is \(\boxed{14}\).

- PM

\(\text{break the figure into 3 pieces}\\ \text{the first piece is 2x1001 white squares to the left of the shaded squares}\\ \text{the second piece is the 2 shaded squares}\\ \text{the third is the 2x1001 white squares to the right of the shaded squares}\\ \text{to pick a rectangle that does not contain the shaded squares we pick two points}\\ \text{from either the first section or the third section and these define our rectangle}\\ \text{there are }\dbinom{1001}{2} =500500\text{ ways to pick 2 points in both sections}\\ \text{so there are }2 \cdot 500500 = 1001000 \text{ ways to pick the rectangles that contain no shaded squares}\)

\(\text{There are }\dbinom{2003}{2} = 2005003 \text{ total ways to pick 2 points from the 2003}\\ \text{thus }\\ P[\text{rectangle contains no shaded square}]= \dfrac{1001000}{2005003 } = \dfrac{1000}{2003}\approx 0.4992\)

yup that is correct.

Let's name X as the number of cookies she has.

Y is the number of cookies she gives to each of her 11 friends, and Z us the number of cookies she gives to each of her 7 friends.

We have X = 11Y + 4 and X = 7Z + 1, and by combining the equations, we have 11Y + 4 = 7Z + 1. Subtracting 4 on both sides, we have 11Y = 7Z - 3, and by dividing both sides by 11, we have \(Y = \dfrac{7Z-3}{11}\). Follow through with the equation.

\(\dfrac{\sum \limits_{k=1}^{1005}~k x^{2k}}{\sum \limits_{j=1}^{1005}~2jx^{2j-1}}=\\ \dfrac{x}{2}\dfrac{\sum \limits_{k=1}^{1005}~2k x^{2k}}{\sum \limits_{j=1}^{1005}~2jx^{2j}}=\\ \dfrac x 2\)

We only need worry about the x values that make the denominator  = 0

This graph shows that there are three of them...

https://www.desmos.com/calculator/hy6vrddzp9

x =  -2     x  =  3    and   x   = 7

1. If x + 2 is a factor, then x = -2 is a root

Using synthetic division

-2  [ 5   8   -7   - 6  ]

          -10   4     6

       ____________

       5    -2  -3    0

Since we get 0 as a remainder, then -2 is a root and (x + 2) is a factor

2.  Likewise....if (x + 1) is a  factor and - 1 is a root.....so...

-1  [   5    8    -7     - 6 ]

              -5   -3      10

        ______________

         5   3    -10     4

Since the remainder is not = 0, then  -1 is not a root  .....so  x + 1  is not a factor

thanks for the answer. :)

Lot the radius be located at (a, 0)

Since the distance from the center to each of these points is the same, we can use the equaion of a circle to find a

(-3 - a)^2  + ( 0 - 2^2 = r^2

(-2 - a)  + ( 0 - 3)^2  = r^2

Since r^2 = r^2.....then

(-3 - a)^2 + (0 -2)^2  =  (-2-a)^2 + (0 -  3)^2       simplify 

a^2 + 6a + 9  + 4  =  a^2 + 4a + 4  + 9      subtract like terms from each side

6a   = 4a

6a - 4a = 0

a(6-4) = 0

2a = 0

a = 0

So...the center is (0,0)

And  the radius is

sqrt [ (-2 - 0)^2 + (0 - 3)^2 ]   =  sqrt [ 2^2 + 3^2]  = sqrt (13)

Thank you, CPhill! Perfect! 

Thank you so much! I understand it now!

(2, 3, 5, 7, 11, 13, 17 )

We have 35 possible sets

(2,3,5) (2,3,7) (2,3,11) (2,3,13) (2,3, 17)

(2, 5, 7) (2, 5,11) (2, 5, 13) (2,5, 17)

(2, 7,11) (2, 7, 13) (2, 7, 17)

(2, 11,13) (2,11, 17)

(2,13, 17)                            

(3, 5, 7) (3, 5,11) (3, 5, 13) (3, 5, 17)

(3, 7, 11) (3, 7, 13) (3, 7, 17)

(3, 11, 13,) (3, 11, 17)

(3, 13, 17)                        

(5, 7, 11) (5, 7, 13) (5, 7, 17)

(5, 11, 13) (5, 11, 17)

(5, 13, 17)   

(7, 11, 13)  (7, 11, 17)

(7, 13, 17)

(11,13, 17)

By the Triangle Inequality....Only 9  are possible

So    9 / 35   

find the slope between the points A(a+b,4m-n) and B(a-b,m+2n) 

(a + b, 4m - n)   = ( X2, Y2)

(a -  b, m + 2n)    =  (X1, Y1)

So.....the slope is

[ (4m -  n) - (m + 2n)]

__________________      simplify

[ (a + b) - (a - b) ]

[ 3m -3n ]

________ 

   2b

3 [ m - n]

_______

   2 b

I would take into consideration the LCM of [7, 11] =77

So, 77n + 15, where n =0, 1, 2, 3.......etc.