CPhill

Let D be the age of Grogg's dad now

Let G ve the age of Grogg seven tears ago

Seven  years ago

(D - 7) = 9G   →   D - 9G  =  7      (1)

Four years ago  (Greg's age =  G + 3 )...so....

(D - 4)  = 6(G + 3)  →  D - 4  = 6G + 18  →  D - 6G  = 22    (2)

Subtract    (1) from (2)

3G  =  15

G = 5

D = 9G + 7

D = 9(5) + 7  =   52

10 members paint  10 sq ft in one minute

So

1 member paints  1 sq ft  in one minute

So

15  paint 15 sq ft / min

So

100 / 15  =  6 + 10/15  =  6 + 2/3  min

x coordinate of vertex =  3  =  -b/ (2a)

So....

6a = -b

b = -6a

Since (0,7)  is on the  graph, then

c = 7

We have 

1  = a(3)^2  - 6a(3)  + 7

-6 = 9a - 18a

-6 = -9a

a = 2/3

b = -6a  =   -6(2/3)  =  -4

a*b*c  = (2/3) (-4) (7)  =  -56 / 3

Slope =  [ 4 -  - 5 ] / [ 5 -  - 1] =  9/6  =  3/2

Equation of the line

y  = (3/2)(x -5 )  +  4        multiply through by 2

2y  =  3(x -5) + 8

2y  = 3x - 7

3x - 2y   = 7          divide through by 7

(3/7)x - (2/7)y  = 1    and we can write

x / (7/3)  - y / (7/2)  = 1

a = 7 /  3

25000 ( 1 - .0825)^15  ≈   6871.18

Possibilities

1 1 1 1 1 6 =   6 different sequences possible

1 1 1 1 2 3  =   6! / 4!  =  30 different sequences possible

36 different sequences

(a+ 2) / (a + 1)  =  b/12

Rearrange as

12 ( a + 2) / ( a + 1)  =  b

This will converge  to ≈  12  as a becomes more positive and  more negative

a         b

0       24

1       18

2       16

3       15

5       14

11     13

-2       0

-3       6

-4       8

-5       9

-7      10

-13    11

x^2 -4x + 5 + x^2 -2x + 1 =  0

(x -1 ) (x + 5) + (x -1)^2  =  0

(x -1) [ ( x + 5) + (x -1) ]  = 0

(x -1) [ 2x + 4]  =  0

x -1  =  0           2x + 4  =  0

x = 1                 x =  -2

Sum  of the roots   =  -B /A  

-2 + 1 =  - B / A

-1  =  -B / A

1 = B/A

A = B

Product of  the Roots = C/A

-2 * 1  = C/A  =1/A

-2 = 1/A

-2A  = 1

A = -1/2

B = -1/2

A + B =    -1/2 + -1/2  =  -1

Call AD  the altitude  =  x

tan B  =   AD / BD =  3/5

tan C = AD / CD =  1/4 =  3/12

BD =  (5)/(5 + 12) (32)  =  160 /17

CD = (12) / (5 +12) (32) = 384/17

So

tan B =  3/5  = AD / BD =  x / BD =    x / (160/17)   →  x = (3/5)(160/17)  = 96/17  =AD

Check this

tan C = 1/4 = AD /CD =  x / CD =  x / (384/17)  → x = (1/4) (384/17) =  96/17 = AD

[ABC] =  (1/2) BC *AD   = (1/2) 32 * (96/17)   = 1536 / 17

xy / (x + y)  =1           xz / (x + z) = 1

xy  = x + y                 xz = x + z

xy - y  = x                  xz - z  = x

y ( x -1) = x               z ( x -1) = x

y = x / (x -1)              z = x / ( x -1)

So   y = z

So

yz / (y + z)  = 1

z*z / ( z + z)  = 1

z^2  = 2z

z^2 - 2z  = 0

z ( z - 2) = 0

z= 0  reject      or     z = 2