geno3141

Draw a line through P that is parallel to BC.

If P is "below" that line, the area of triangle(PBC) will be less than one-half the area of triangle(ABC).

The area below that line is three times the area above the line, so the probability that the area of

triangle(PBC) is less than the area of triangle(ACB) is 3/4.

The probability that the first occurs, the second doesn't and the third doesn't:  (1/3)(5/7)(4/5)

or

the probability that the first doesn't occur, the second does and the third doesn't:  (2/3)(2/7)(4/5)

or 

the probability that the first doesn't occur, the second doesn't and the third does:  (2/3)(5/7)(1/5)

Find each of the above individual probabilities and add their answers together.

For BD to bisect angle(B) and also to bisect the side AC, triangle(ABC) must be an isosceles triangle with

AB = BC.

Since AB = AC, all three sides, AB, BC, and AC are equal in length.

This means that triangle(ABC) is an equilateral triangle, so each of its angles must equal 60^o.

If it takes 12 teenagers to eat one pizza in 20 minutes, they spent  12 x 20  =  240  teenager-minutes chewing.

So, how many teenagers will it take to eat that pizza in 18 minutes?

Divide the number of teenager-minutes by minutes to get the number of teenagers.

240 teenager-minutes  /  18 minutes  =  13 1/3 teenagers.

There are four prime numbers (2, 3, 5, 7) in the first 10 whole numbers.

The probability of rolling a prime is  4/10  and the probability of not rolling a prime is  6/10.

The probability that exactly three of the dice show a prime number is:

                      ₄C₃ · (4/10)³ · (6/10)¹

If the legs have values  x  and  y     its area is  ½·x·y     and the sum of its legs is  x + y

The area is equal to three times the sum of its legs:     ½·x·y  =  3(x + y)

Solving:                                                                                xy  =  6(x + y)

                                                                                            xy  =  6x + 6y

                                                                                      xy - 6y  =  6x

                                                                                     y(x - 6)  =  6x

                                                                                              y  =  6x / (x - 6)

Substituting values for x to get integral values for y gives:  

     (7, 42)     (8, 24)     (9, 18)     (10, 15)     (12, 12)     (15, 10)     (18, 9)     (24, 8)     (42, 7)

Removing the duplicates:     (7, 42)     (8, 24)     (9, 18)     (10, 15)     (12, 12) 

Finding the areas of these triangles and summing their answers will give the final answer.

The formula for the sum of a finite geometric series is:  Sum  =  [ a · ( 1 - rⁿ ) ] / (1 - r)

For this problem, the first term  a = 1/4.

For this problem, the common ratio  r = 1/2.

So:  15/32  =   [ ¼ · ( 1 - ½ⁿ ) ] / (1 - ½)

        15/32  =   [ ¼ · ( 1 - ½ⁿ ) ] / (½)

        15/32  =  ½ · ( 1 - ½ⁿ )

        15/16  =  1 - ½ⁿ

         -1/16  =  - ½ⁿ 

           1/16  =  ½ⁿ 

                n  =  4

The decimal fraction is:  0.515 515 515 515 ...

Let             x    =    0 . 515 515 515 515  ...

Then  1000x  =  515 . 515 515 515 515 ...

Subtracting the first line from the second:

              999x  =  515

                    x  =  515 / 999

The probability that the first cube has red facing upwards is  3/6  =  1/2.

Similarly, the probability that the second cube has red facing upwards is  3/6  =  1/2.

The probability that both cubes have red facing upwards is  1/2 x 1/2  =  1/4.

The first term of an arithmetic progression is  4.

The common difference is  d.

The fifth term is  4 + 4d.

The sum of the first five terms is:  Sum  =  5[ 4 + (4 + 4d) ] / 2  =  20 + 10d

The sixth term is  4 + 5d

The tenth term is  4 + 9d

The sum of the next five terms is:  Sum  =  5[ (4 + 5d) + (4 + 9d) ] / 2  =  20 + 35d

Since the sum of the first five terms is one-fourth the sum of the next five terms:

              20 + 10d  =  ¼( 20 + 35d)

Solving:  d  =  -12,'

Since you know the first term and the common difference, you can find the 15th term.