CPhill

OK, EP

I'll  "unblock" yours, as  well !!!

This is  crazy  !!!

Area /  circumference =   1/2

(pi r^2) / (2 pi r)  =  1/2

r / 2 = 1/2

r =  1 cm

a. Rate =  cups / hrs

So

A =  160 / 1   = 160  cups per  hour

B =  560 / (2 1/3)  =  560 / ( 7/3)  =  560 (3/7) =  240 cups per hr

b.

1800 / [ 160 + 240] =  1800 /  400 =  18/4 = 9/2  =  4 .5 hrs  

I don't know why!!!

I just returned  to this site in the last few days and some strange things are  going  on.....anyway....I "unblocked" your answer !!!!

1/ (2x)  = ( r - x) / 2         cross-multiply

2 = 2x ( r - x)

2 = 2xr -2x^2       rearrange as

2x^2 -2xr + 2 = 0    simplify

x^2 - xr + 1  =   0

This will  have  one real solution  when

r^2 - 4(1)(1) =  0

r^2  - 4  = 0

r^2 = 4             take both  roots

r = 2 , -2

And the  product of these =  -4

-3n + 3 < -11 -  4n

-3n + 4n < -11 -3

n < -14

No positive integers are  solutions

Thx, GA.....a lot of  "strange stuff" going on  here than before......I long for the old days when I only had to  deal  with  some "regular" morons......these morons are much  more  quasi-sophisiticated....

log _√3 3^4 = log_x 4

4 log _√3 3 =  log _x 4

4 log _√3 (√3)^2  = log _x  4       

(2 * 4 )  log _√3 (√3)   = log _x 4              (    log _√3 (√3) =  1)

2 * 4 * 1    = log _x  4

8  = log _x  4           (x must be positive)

This means that    x^8 = 4    →   take the positive 8th root of both sides....

x  = 4^(1/8) =  (2^2)^(1/8) = 2^(1/4)

So

log ₂  x  =   log ₂ 2 ^(1/4)  =    (1/4) log ₂ 2    = (1/4) * 1  = 1/4

And you are  correct !!!!!

We have a geometric series  where the  ratio between successive terms  =  r

The sum of such a series, S, can be represented by

S =   first term / ( 1 - r)   .....so we have.......  

8 =10 / (1 - r)

1 - r = 10/8

r = 1 - 10/8

r =  -2/8  =  -1/4

See if this helps :

https://www.mathhomeworkanswers.org/265744/help-help