CPhill

13

14  could be arranged as 7 x 2

15 could be arranged as 5 x 3

13 can only be arranged as 13 x 1

a3  = a1 + 2d

a2 = a1 + d

a4 = a1 + 3d

a6 = a1 + 5d

So we have these equations

a1 + a1 + 2d = -2

a1 +d + a1 + 3d + a1 + 5d = 1     simplify

2a1 + d = -2

3a1 + 9d = 1

Multiply the first equation by -9  and add to the  second

-15a1 = 19

a1  =  -19 / 15

This is impossible

Leg AC  would be >  6  while  AB (the hypotenuse)  would only  = 1

x^2 + 5y^2  = 30

Take the derivative with respect to x and set to 0

2x = 0

Take the derivative with respect ot y and  set to 0

10y = 0

This implies that

2x =10y

x = 5y

So

(5y)^2 + 5y^2   =30

30y^2  = 30

y = 1

x = 5y  =  5

Mav  x + y   =  6

A                         B

                           4

                           P

                           1                          

D    2     Q    3    C

PA =  sqrt [ 5^2 + 4^2] = sqrt 41

PQ = sqrt [ 3^2 + 1^2] = sqrt 10

AQ = sqrt [ 5^2 + 2^2]  = sqrt 29

Law of Cosines

PQ^2 = PA^2 + AQ^2 - 2(PA * AQ) cos (PAQ)

10 = 41 + 29  - 2 ( sqrt [41 * 29] ) cos (PAQ)

cos (PAQ)  =  (10 - 41 - 29 ) /  (-2 *sqrt [ 41 * 29 ] )  =  30 / sqrt 1189

sin (PAQ)  =  sqrt [1189  - 30^2 ] / sqrt 1189   = sqrt [289] / sqrt 1189  =  17 / sqrt 1189

Simplify as

x^2 + 12x + y^2 + 2y  =  10        complete the square on x, y

x^2 + 12x + 36 + y^2 + 2y + 1 =  10 + 36 + 1

r^2  =  10 + 36 + 1   = 47

Area  = r^2 * pi =  47 pi ≈ 147.65

a)  3 (8/10)  + 8 (2/10)  =  24/10 + 16/10  =  40 / 10   =  4 

b) 3(8/11) + 8(3/11)  =  24/11 + 24/11  = 48 / 11  ≈  4.36

c) 3 (8/12) + 8 (4/12)  =   24/12 + 32/12  =  56 / 12 =  14 / 3 ≈ 4.67 

d)  3 [ 8 / (10 + n) ]  +  8  [ (2 + n) / (10 + n)  ]  = 6

[ 24 + 16 + 8n ] / [10 + n ]  =  6

40 + 8n  = 6 [ 10 + n]

40 + 8n  = 60 + 6n

8n - 6n =  60 - 40

2n = 20

n = 10

e)   3 [ 8 / (10 + n) ]  +  8  [ (2 + n) / (10 + n)  ]   ≥ 7

40 + 8n   ≥  7 [ 10 + n ]

40 + 8n  ≥ 70 + 7n

8n - 7n  ≥ 70 - 40

n  ≥ 30

x^2 -10x + y^2 + 10y  =  -c     complete the square on x, y

x^2 -10x + 25 + y^2 + 10y + 25  = -c + 25 + 25

-c + 25 + 25   =  1

-c + 50  =1

-c = -49

c = 49

Let x be the smallest angle

Let  x + 2d   = the largest

(x + 2d)  - x  = 56

2d = 56

d =28

So

x + (x + d)  + (x + 2d)   =180

3x + 3d   =180

x + d  = 60

x + 28 = 60

x =60 - 28 =  32  = the smallest angle

We will have 6 congruent triangles with sides of 3 and an included angle of 60°

Area  = 6 (1/2) (3^2) * sin 60  =

6 (1/2) (3^2) * sqrt (3)  / 2

27 * sqrt (3) / 2  =

13.5 *sqrt (3)  ≈ 23.4