CPhill

Let the amount of Alice's original money = A

Let the amount of Bob's original money = B

So

(A + n )* 3  =  (B - n)     → 3A + 3n  = B - n  →  3A - B  = -4n   (1)

(A - n) =  2 * ( B + n)  →  A - n  =  2B + 2n →  A - 2B  = 3n  (2)

Multiply the first equation by 3  and the second equation by 4  and add them together

9A -  3B  =  -12n

4A  - 8B =    12n

13A  - 11B  =  0     add 11B to both sides

13A  = 11B          divide both sides by 13, B

A / B  =  11 / 13

 At the Dessert Island Pastry Cafe, there are 2 cakes for every 7 donuts and 11 brownies for every 3 cakes. What is the ratio of brownies to donuts?

2 cakes : 7 donuts      →   6 cakes : 21 donuts

3 cakes : 11 brownies  →  6 cakes : 22 brownies

22 brownies : 21 donuts

ax^4-bx^2+x+5$ and $f(-3)=2

a(-3)^4 - b(-3)^2 + -3 + 5  = 2

81a - 9b +  2  = 2

81a - 9b  = 0

9a  =  b

So  f(3) = 

a(3)^4 - 9a(3)^2+ 3 + 5 =

81a - 81a + 3 + 5

3 + 5   =  

8

The minute hand has moved 30° from 12 o'clock

The hour hand at 3 o'clock  is 90° from 12 o'clock

And It moves 30° every hour....so...in 5 minutes....it would have moved  (1/12) of 30° = [30/12]° =

(5/2)°  = 2.5°

So......the  [lesser]  angle  between the minute and hour hand at 3:05  is

[ 90 + 2.5] -  30  =  62.5°

2x^2 + 4x - 7 = 0

Add 7 to both sides

2x^2 + 4x  =   7       divide through by 2

x^2 + 2x  = 7/2 

Take (1/2) of the coefficient on x  = (1/2)(2)  = 1

Square this  = 1   and add it to both sides

x^2 + 2x + 1  =  7/2 + 1        factor the left and simpify the right

(x + 1)^2  =   9/2                  take both roots

x + 1  =   ±√ [9/2]

x + 1  = ±3 /√2        rationalize the denominator on the righr

x + 1  =  ±3√2  / 2      subtract 1 from both sides

x =  ±3√2  / 2  -  1

x = ±3√2  / 2  - 2/2

x = [ -2  ± 3√2 ]   / 2

The radius of the cylinder = 30mm

And its length = 240 mm

So....the lateral surface area is    2pi * radius  * length  = 

2pi * 30 * 240  =  14400 pi mm^2

And the surface area of the two ends is   pi * 30^2  = 900pi mm^2

So the total surface area of the cylinder  is  [ 14400+ 900 ]pi mm^2  =  15300 pi mm^2  ≈ 

48,066 mm^2

The total surface area of the box is

[2* 120^2 + 4*120*60 ]  =  57,600 mm^2  

So the cylinder has the smaller surface area

The radius of the sem-circle is 12mm

And its area =  (1/2) pi * 12^2  =  72 pi

The radius of the small circle is 6mm   and its area  =  pi * 6^2  =  36 pi

So....the shaded area is the difference of the two  =  [72 - 36] pi  = 36 pi ≈  113 mm^2

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

3x^2 - 3x  =  0         this shifts the graph of  3x^2 - 3x -1  up one unit

So   3x^2 - 3x  =  0   at x = 0  and x = 1

3x^2 - 3x -1  =  x + 1

The graph of x + 1  would intersect 3x^2 - 3x -1   at  x ≈ -4/10   and  x ≈ 1.5

b)  iii   .the "blue" curve is the correct one.......when  y = 1.......x ≈  -1 + 1/4  = - 5/4  and  x ≈ 3 + 1/4  =  13/4

Here's the first one:

4x^2 - 4x  - 1  = 0    two solutions appear to be x  = -1/4   and x =  5/4

4x^2 - 4x  - 1  = 2    two solutions appear to be x  = -1/2   and x =  3/2

There are 12 possible outcomes.....three of these will be a "7"  ...so...

 A "7" should occur  1/4 of the time

If each player pays 20p to play.....the school  will take in  120 * 20  = 2400p

However.......1/4 of the players should win =  30 players.....and each lollipop costs 30p.....so......the school should spend 30 * 30  = 900p for the prizes

So......the school should raise   2400 - 900 = 1500p  if 120 people play the game