CPhill

f(g(x)  =  2 [x^2] + 1

g(f(x))  =  [ 2x + 1 ] ^2

So

f(g(x))  =  g (f (x) ) 

2 [ x^2]  + 1  =  [2x +  1 ] ^2

2x^2 + 1  =  4x^2 + 4x + 1      subtract  2x^2 , 1  from both sides

2x^2 + 4x  = 0      factor

2x ( x  + 2 )  = 0

Setting each factor to 0  abd solving for x produces  x = 0 , -2

f(g(x)  =  [2x^2 + 3 ]  - 1

g (f(x))  = 2 [ x - 1 ] ^2  + 3  

So.....we are trying to solve

f(g(x)  = g(f(x))

[ 2x^2 + 3 ] - 1  =  2 [ x - 1 ]^2 + 3     simplify

2x^2  +  2   =   2 ( x^2 - 2x + 1) + 3

2x^2 + 2  = 2 x^2 - 4x + 2 + 3     subtract 2x^2 from both sides

2  =  -4x + 5   add 4x to both sides, subtract 2 from each side

4x  =  3   divide both sides by 4

x  = 3 / 4

We can answer this with a Venn Diagram

There are  30 total students

The total that play  Football, Cricket or Both  = 26

So.....the probability  is      26 / 30   =     13 / 15

a. cos (60)  =   1/2

b. Using the Law of Cosines

b^2  =  8^2  + 5^2  -   2 (8) (5) cos (1/2)

b^2  =  64 + 25  - 80 (1/2)

b^2   =  89  -  40

b ^2  =  49       take the positive root

b  = 7

4)Sixteen times the reciprocal of the number of inches in the circumference of a circle equals the number of inches in the diameter of the circle. What is the number of square inches in the area of the circle?

I like this one.....

Call the number of inches on the circumference,   I

So

16/ I  = D  =  2r    ⇒  16/ I  =  2r   ⇒   I   =  16/2r   ⇒   I  =  8/r

So.......

Circumference in inches =  2 * pi * r

8/r  = 2 * pi * r         multiply both sides by r

8  = 2 * pi * r^2       divide both sides by  2 pi

8 /  [ 2pi ]  =  r^2

4/pi  = r^2

So....the area  is

pi  (r^2)  =

pi  ( 4/pi)   =     4 units^2

3)A sector of a circle has a central angle of 80 degrees. If the area of the sector is 32pi, what is the radius of the circle?

Area  =    pi  * radius^2  ( 80 / 360)  =

32 pi  =   pi  * radius^2  ( 2/9)           divide out pi

32   =   (2/9)  * radius^2       multiply both sides by 9/2

32 (9/2)  =  r^2

144  = r^2         take the positive root of both sides

12  = r

Thanks, lynx  !!!!

1)What is the length of a 100 degree arc in a circle of radius 9?

We have that

Arc Length  =  2 pi radius *   ( 100/ 360)  =   2 pi * 9  * (10/36)  =   18 pi * (5/18)  =

5 pi   units

I liked it, too....I'll have to say that the answer isn't exactly intuitive...it would seem that it would take him more time to turn around and retrieve the hat than the time elapsed between when he lost it and when he realizes that he lost it....but....nope  !!!