CPhill

OK  !!!

The idea is to get x  by itself

y  =  (1/2)x + 1      subtract 1 from both sides

y - 1   =   (1/2)x       multiply both sides by 2

2(y - 1)  =    x

Correct   !!!

Area  of  pizza  with 12 in diameter  =  pi (12//2)^2  =  pi (6)^2  =   36pi  in^2

So each person eats  18pi in ^2

Then  4 people  must  eat    4 (18 pi)  =  72 pi  in^2

So....we need to solve this

72pi  =  3 (pi * r^2)     divide both sides by 3 pi

24  = r^2   take the positive root

sqrt (24)  = r

So....the diameter =  2r  =  2qrt (24)  = 2 *2sqrt (6)  =   4sqrt (6)  in 

No prob  !!!!

The first  summation implies  that

[a1 + a21] [21/2]  = 693

[ a1 + a21] [10.5]  = 693       divide both sides  by  10.5

a1  + a21  =  66

So   we have 10 pairs of terms  each summing to  66

So....the 11th term must be   693 - 10(66)  =  693 - 660 = 33

And the  second summation  sums  the   odd tems  of  the series

So....

(a1 + a21)  =  66

(a3 + a19) =   66

(a5 +  a17) =  66

(a7 + a15) =   66

(a9 + a13) =   66

  a11         =   33

So....the  second sum  =   5(66) + 33   =   363

Let N be the  number of books she started with 

To Fred she sells  2 books  and then 1/5 of what is left  =    2 + (1/5)(N - 2)    books

To Joan....she sells  6  books and what remains

What remains  is  what Verna started with  - what she sells to Fred  - 6 books she sold to Joan

So she sells Joan  6 + (1/5)(N - [2 + (1/5)(N-2)]  - 6)

So  we have this inequality

2 + (1/5)(N - 2)  >  6 + (1/5)(N - (2 + (1/5)(N - 2)) - 6)    mutiply through by 5

10  + (N - 2)  >  30 +  (N - (2 + (1/5)(N - 2)) - 6)

N + 8  >   30 + ( N - (2 + (1/5)N - 2/5) - 6)

N + 8 > 30 + (N  - 2 - (1/5)N  + 2/5  - 6)

N + 8 >  30  + (4/5)N - 38/5      multiply through by 5

5N + 40  > 150 + 4N - 38

5N + 40 > 150 + 4N  - 38

N > 150 - 38 - 40

N > 72

Then...she had  to  have at  least 73 books in her collection

Check....if she started with just 72 books

She sells  to Fred    2 + (1/5)(72 - 2)  =  2 + 14  =  16 books

She has  56 books left

She sels to Joan   6  + (1/5) (56 -6)  =  6 + 10 = 16 books

But....she sells to Fred more books.....so....N must be greater than 72

(t^2 - 82)^2  =  324     expand

t^4  - 164t^2  + 6724  = 324    subtract 324 from borh sides

t^4 - 164t^2  +  6400   =   0     we can factor this as

( t^2  - 100) ( t^2 - 64)  =  0    factor these terms  as differences of squares

(t-10)(t+10)(t-8)(t+8)  =0

Setting each factor to 0  and solving for t  produces  the solutions

t  =10   t  = -10    t  =  8    t  = -8

2x + 3y  = 100

3y  =  100 - 2x

y  =  100 - 2x

       _______

             3

We can only have integer solutions for y  whenever  x  = 2 , 5, 8, 11, etc

And the last  x  that  makes y a positive integer  is  when x  = 47

So....the  number  of  integers  =   [ 47 - 2 ] / 3  + 1   =   45 / 3 + 1   =   15  + 1  =  16

Just as Lucky Ducky found   !!!!!

GOOD JOB LUCKYDUCKY  !!!!!

Let d  be  the  original distance to the building

Let h be the height of the building

Since the original  angle  of elevation is 45°, then d  =  h

So  when we move 20 feet back  we have that

tan 30°   =   d   / [d + 20]      rearrange as

[ d + 20] tan30°  =  d

d tan 30 + 20 tan 30°  = d

  20 tan 30°  =  d - dtan30°

20tan30° =  d ( 1 - tan 30°)

[ 20tan30° ] / [ 1 - tan 30°]  =  d  ≈ 27.32 ft  =  h