CPhill

y(1+x^2)y'-x(1+y^2) = 0     this is a separable equation

y ( 1 + x^2)y'  =  x ( 1 + y^2)   rearrange as

[ y] / [ 1 + y^2 ] y'  =  [x] / [ 1 + x^2]   integrate both sides

∫ y / [ 1 + y^2] dy  = ∫ x / [ 1 + x^2]  dx

(1/2) ln ( 1 + y^2)  =   (1/2) ln ( 1 + x^2) + C

ln (1 + y^2)  =  ln ( 1 + x^2 ) + C    exponentiate both sides

e^[ln (1 + y^2) ] =  e ^ [  ln ( 1 + x^2 ) + C]

1 + y^2  =  e^[ ln ( 1 + x^2)] * e^C       →     let e^C   =  C₂

1 + y^2  =  (1 + x^2) * C₂

y^2  = ( 1 + x^2) *C₂  - 1

y = ± √  [   ( 1 + x^2) *C₂  - 1 ]

y (x) =   ± √  [   C₂x^2 + C₂  - 1 ]

Now....since the intial condition is that   y(0)  = √3  .....the positive root must be correct....solving for C₂, we have that

√3  =   √  [   C₂(0)^2 + C₂  - 1 ]

√3 = √ [ C₂ - 1]

3  = C₂ - 1

C₂ = 4

So....the particular solution is   y(x)  = √  [   4x^2 + 4  - 1 ]  =   √  [  4x^2 + 3 ]

Here's another way to determine t  with polynomial long division

                                 4x^2  -  6x  + 10

5x^2 - 6x + 7     [     20x^4 -  54x^3  + 114x^2  - 102x  + 70  ] 

                                20x^4 - 24x^3  +    28x^2

                                ________________________________

                                             - 30x^3 +  86x^2  - 102x   

                                            - 30x^3  + 36x^2  -    42x

                                            _____________________  

                                                            50x^2  -  60x  + 70

                                                            50x^2 -  60x   + 70

                                                           ___________________

So...it appears that  t =  -6

Assuming that the side lengths correspond.....note that the ratio of the first two sides in each triangle = 18 / 12  = 3 / 2

So.....x   will  =  (3/2) 14 =   21 cm

If they from an arithmetic progression...let the smallest angle  = x    and the largest angle  = x + 44

So...we have that

x +  4d  =  x + 44      where d is the common differnce between consecutive angles

4d  = 44

d  = 11

And the interior angles of a pentagon will sum to 540°....therefore

x + (x + 11) + (x + 22) + ( x + 33) + (x + 44)  = 540    simplify

5x + 110  = 540    subtract 110 from each side

5x = 430      divide both sides by 5

x = 86°   and this is the smallest angle

The other angles are   97°, 108°, 119° and 130°

And [ 86 + 97 + 108 + 119 + 130 ] =   540

Here's another way without using vectors.....let B  = (x, 0)

If the right angle is at A.....the hypotenuse is BC....and this distance  is  just  

sqrt [ (7 - x)^2 + (3 - 0)^2 ] =

sqrt [ 49 - 14x + x^2 + 9 ]   =  sqt [ 58 - 14x + x^2 ]

And AB  forms one of the legs...and its length is just    sqrt [ (-2 - x)^2 + (6 -0)^2 ] =  

sqrt [ ( 4 + 4x + x^2 + 36 ] =

sqrt [ 40 + 4x + x^2 ]

And AC  is the other leg....and its  length is just  sqrt [ (-2 - 7)^2 + (6 -3)^2 ] =

sqrt [ 81 + 9 ]  = sqrt [90]

And by the Pytahgorean Theorem.........AB^2 + AC^2  = BC^2   ....so...

[ 40 + 4x + x^2 ]  +  90    =  [ 58 - 14x + x^2 ]   simplify

130 + 4x  =  58 - 14x          subtract 4x, 58 from both sides

72 = -18x             divide both sides by -18

-4  = x       so ....B  = (-4, 0)

Obviously.....vectors make the process easier.....!!!!

Area of a sector  =  theta * pi * r^2 / 360   .....so...we have

30  =  theta * pi * (225) /360  simplify

30  = theta  * pi  * (5/8)      divide both sides by   pi * (5/8)

30 / [ pi * 5/8]  = theta  ≈  15.3° 

Yep....that makes it way easier.....!!!!

Excellent thinking, hectictar....!!!!!....this one was a little tricky, for sure....!!!!!

Here's one more thought......dropping a perpendicular from Z to H on  AC means that  ZH is parallel to BA....and whenevever   a segment is drawn parallel to a base, it splits the sides of the  triangle into  equal ratios....that is.....CZ / BZ  =  CH / AH.....but since BZ  = CZ  [ because Z is the midpoint of BC ], then  AH  = CH

Then by SAS, triangle CHZ   is congruent to triangle AHZ.....and since you found that BCA  = 32°  = ZCH  =  ZAH  = ZAC

Obviously.....dropping that perpendicular as you did was the key to the whole thing.....not bad for a 'Bama fan....LOL!!!!

Sonji has $5.25 in nickels and dimes.The coin value can be modeled by the equation 0.10d + 0.05n = 5.25, where d represents the number of dimes and n represents the number of nickels. If Sonji has 21 dimes, how many nickels does she have?

.10 (21)  + .05 n  = 5.25

2.10  + .05n  = 5.25   subtract  2.10 from both sides

.05n  = 3.15           divide both sides by .05

n  = 3.15 / .05    =    315 / 5  =  63  nickels

Here's another way :

The first and last terms sum to 26....so....the sum is given by

[ Sum of first and last term] [ Indices of last term  - indices of first term + 1 ]  / 2  =

[26 ] [7 - 3 + 1 ] / 2  =

[26] [ 5] / 2  =

[26 / 2 ] * 5   =

13 * 5  =

65