CPhill

                                 A

                            10         15

                  B      1     D       3                  C

AD^2  =    AB^2  - (BC/4)^2

AD =   AC^2  - [(3/4)BC]^2 

So

AB^2  - (BC/4)^2  =  AC^2  - [(3/4)BC]^2

10^2  - BC^2 / 16  = 15^2  - (9/16)BC^2

15^2  - 10^2  =  (9/16 - 1/16)BC^2

125  = (1/2)BC^2

250  = BC^2

AD^2  =  10^2  - (BC/4)^2

AD^2  = 100 -  (1/16)BC^2

AD^2   =100  -  250/16

AD^2   =    675 / 8

AD  = sqrt (675 / 8)   = (15/4)sqrt 6  ≈ 9.19

A

B               C

AC =  sqrt [ (4sqrt 2)^2 + 5^2  ] =  sqrt  [ (sqrt (32))^2 + 5^2 ] =  sqrt (57)

cos (ACB)  =  5/sqrt (57)

Law of Cosines

BM^2   = (AC / 2)^2  + BC^2  - 2 ( AC/2 * BC) (5/ sqrt 57)

BM^2  = AC^2 /  4  + BC^2  - (AC*BC) ( 5/sqrt 57)

BM^2  =   57 / 4  + 25 - (sqrt (57) * 5) (5  /sqrt 57)

BM^2  = 57/4 + 25 - 25

BM  =  sqrt (57)  /  2 =  AC / 2

\(6x^2 + \frac{3}{5}y^2 - 2x + 8y \)

xy = 2/5  →  y = 2 / (5x)

6x^2  + (3/5)(2 / (5x)^2 - 2x + 8(2 / (5x))

6x^2  + (12/125)x^(-2) - 2x + (16/5)x^(-1)

The graph shows that the minimum ≈ -26.525  when  x ≈ -.06

\(f(x) = \frac{(2x + 3)(x - 7)}{\sqrt{x}} \)

The denominator  cannot  =  0 

In the radical, x  must be >  0

Domain  = (0 , infinity)

1-9  = 9 digits

10 - 99 = 180 digits

100 - 199 = 300 digits

200 - 250  = 151 digits

9 + 180 + 300 + 151 =     640 digits

No. of 2's

1-100  =  10 + 10  = 20

101 - 199 = 10 + 10  = 20

200 - 250  = 5 + 10 + 51  = 66

20 + 20 + 66  =  106

Prob of 2  =   106  / 640 =  53 /  320    (if I didn't mis-count !!! )

5n / (n + 2)  =  6

Can you solve this ???

(n - 3) + (n - 2) + (n -1) + n + (n + 1) + (n + 2) + (n + 3)  = 7(n + 3)

7n = 7(n + 3)

7n = 7n + 21

No solution

Triangle COB is similar to Triangle DOB  by HL

tan (angle  COB) =   1/sqrt(3) = sqrt (3) / 3   = tan angle DOB

tan angle DBC = tan angle ABC

tan (angle ABC)  = tan (angle COB + angle DOB)  = tan (2 angle COB) = AC / BC

tan (angle COB + angle DOB )  =   [ tan COB + tan DOB ] / [ 1 - tan COB*tan DOB] =

[ 2sqrt (3)/3 ] / [ 1 - (sqrt (3) / 3)^2 ] =  (2sqrt (3)/3) / [ 1 - 1/3]  = (2sqrt(3)/3) / (2/3)  =

sqrt 3  =  AC / BC

sqrt 3  = AC / sqrt 3

sqrt 3 * sqrt 3  = AC  =  3

25 min =   5/12 hr

Call your  normal time at 40 mph = T

So    rate * time =  distance

Equating  distances

40T = 55 (T - 5/12)

40T =  55T  - 275/12

275 /12  =  (55 - 40)T

275 / 12  = 15T

T = 275/(15*12)   = 55/36 hr

Miles driven =   40 (55/36)  ≈  61.1 miles