CPhill

If the discriminant  =  0.....we have one  solution

If the discriminant > 0,  then we have two solutions

If the discriminant  is < 0, we have two non-real solutions

Let B   = L

Let A = L - 11

Let C  = (1/2)B    = (1/2)L  = .5L

So we have that

A + B + C   =  29

(L -11) + L + (.5)L  = 29         simplify

2.5L -  11  = 29             add 11 to both sides

2.5L =  40                     divide both sides by 2.5

L  = 16

A =  L - 11  =   16  - 11   =    5 cm

At least 8 means that   either 8, 9 or 10  heads were tossed

So  we have

C(10, 8)  + C(10,9)  + C(10,10)  =

45   +  10   +     1   =

56 possible sequences

Second one....

Yep.....  r   = 1/7    and it converges  

Your answer to the first one is good    !!!

First one

The 4th term in the expansion  of     [ (2x^2y^3)  +  y ]^7   is given by

C(7, 3) * (2x^2y^3)^(7 - 3) * ( y)^3  =

C(7,3) * ( 2x^2y^3)^4 * y^3   =

35 * 2^4  *  (x^2 y^3)^4 * y^3  =

35 * 16  *  x^8 * y^12 * y^3  =

560 x⁸ y¹⁵

Here's the complete expansion :

128 x^14 y^21 + 448 x^12 y^19 + 672 x^10 y^17 + 560 x^8 y^15 + 280 x^6 y^13 + 84 x^4 y^11 + 14 x^2 y^9 + y^7

*****

49.13   - 2(32.50)  + 2(74.25)    = 

49.13  -  65  + 148.50  = 

$132.63 ending balance

See the following image

Note that OXQY  forms a quadrilateral

Since  QX and QY   are tangent to the circle, then angles  QXO  and QYO   = 90 °

And by symmetry,  angle   XQY  = 2(32)    = 64°

And the interior angles of a quadrilateral sum to 360°

So

QXO + QYO + XQY  + XOY  =  360

  90  +   90   +   64  + XOY  =  360

244 + XQY  = 360         subtract  244 from both sides

XOY  = 116°

And since  XOY  is a central angle........then minor arc  XY has the same measure  =   116°

195 + 10  ≤  40n  

In 60 seconds, the ostrich will run  60*44   = 2640 ft

And let  S  be the number of seconds that it will take the cheetah to catch the ostrich

So....the distance that  the ostrich will cover before being caught  is  2640 + 44S

So....we need to solve this

110S  = 2640 + 44S        subtract  44S from both sides

66S  =  2640      divide both sides by 66

S = 40 seconds