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1.A committee of three people is to be randomly selected from a group of three men and two women, and the chairperson will be randomly selected from the committee. What is the probability that the committee will have exactly two women and one man, and that the chairperson will be a woman? Express your answer as a common fraction. 

 

2.A group of 25 friends were discussing a large positive integer. "It can be divided by 1," said the first friend. It can be divided by to said the 2nd. and by 3 said the 3rd ... This continued until and by 25. If 2 of these friends are incorrect, what was the least possible integer they were discussing? And you do not take out 24 ann 25, I tried that!!!

 

3.The total number of degrees in the sum of the interior angles of two regular polygons is 1980. The sum of the numbers of diagonals in the two polygons is 34. What is the the positive difference between the numbers of sides of the two polygons?

 

4. A triangle is formed by connecting the points (-5,0), (0,6), and (5,0). The resulting triangle is then rotated about the x-axis to form a solid. The original triangle is then rotated about the y-axis to form a different solid. What is the positive difference between the number of cubic units in the volumes of the two resulting solids? Express your answer in terms of pi.

MIRB16  Feb 26, 2018
edited by MIRB16  Feb 26, 2018
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 #1
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3.The total number of degrees in the sum of the interior angles of two regular polygons is 1980. The sum of the numbers of diagonals in the two polygons is 34. What is the the positive difference between the numbers of sides of the two polygons?

 

Call the number of sides of the first polygon, a   and the number of sides of the second polygon, b

 

The sum of the interior angles is given by

 

[ (a - 2) + (b - 2)]  * 180  =  1980

[ a + b] - 4    =  11

[ a + b ]  =  15

b  =  15 - a

 

The number of diagonals in a polygon  =  n(n - 3) / 2    where n is the number of sides

 

So....the sum of the diagonals is

 

(a) (a - 3) / 2   + (b)(b - 3)/ 2  =  34

a^2 - 3a + b^2 - 3b  =  68

a^2 + b^2 - 3(a + b)  = 68

a^2 + b^2  - 3(15)  = 68

a^2 + b^2  =  113

a^2 + (15 - a)^2  = 113

a^2 + a^2  - 30a  + 225  = 113

2a^2 - 30a  + 112  = 0

a^2 - 15a + 56  =   0

(a - 7) (a - 8)  =  0

 

So...a  = 7 , b = 8  or a  = 8, b = 7

 

So....the positive difference between their number of sides  =1

 

cool cool cool

CPhill  Feb 26, 2018
 #2
avatar+86536 
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4. A triangle is formed by connecting the points (-5,0), (0,6), and (5,0). The resulting triangle is then rotated about the x-axis to form a solid. The original triangle is then rotated about the y-axis to form a different solid. What is the positive difference between the number of cubic units in the volumes of the two resulting solids? Express your answer in terms of pi.

 

The first object needs a full rotation about x to form a solid and it will form two cones....each with a radius  of 6  and a height of 5

So....their volume will be ;

(2/3)pi (6)^2 (5)  =  120 pi  unit's^3

 

The second object only needs a half rotation around y to form a solid and it will form a cone with a radius if 5 and a height of 6

So....its volume will be :

(1/3) pi (5^2) (6)  =  50 pi units^3

 

So...the diference between their volumes is    70 pi  units^3

 

 

cool cool cool

CPhill  Feb 26, 2018
 #3
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2)  25! =1.5511210043330985984e+25 = 2^22 * 3^10 * 5^6 * 7^3 * 11^2 * 13 * 17 * 19 * 23 / 2^18 * 3^8 * 5^4 * 7^2 * 11 =26,771,144,400= 2^4 * 3^2 * 5^2 * 7 * 11 * 13 * 17 * 19 * 23. This number of 26,771,144,400 will still divide all numbers from 1 to 25. But if 2 of them were wrong, then we can divide it by the 2 larget primes under 25, that is: 19 x 23. So, 26,771,144,400 / [19 x 23] =61,261,200.

This last number of 61,261,200 will still divide every number from 1 to 25 EXCEPT 19 and 23.

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