CPhill

Heureka's method is from Euclid.....

It says that, if a - b = c  and c / b, then c / a   and c is the greatest common factor of a and b

So

   1135620

 - 1134218

-------------

        1420 

And 1420 / 1134218  →    so,  1420 / 1135620     and is the greatest common factor of both numbers......

Asumiing we have a function

N = ab^t    where N is the bacteria count at some time t........note.....when t  = 10, N = 800....and when t = 40, N = 1800...so we have

800 = ab¹⁰     and solving for "a," we have     800/b¹⁰ = a

And we also have

1800 = ab⁴⁰      and substituting, 800/b¹⁰  for a, we have

1800 = (800/b¹⁰)b⁴⁰ →   1800 = 800b³⁰

And dividing both sides by 800  we have

1800/800 = b³⁰  →  9/4 =  b³⁰

And taking the (positive) 30th root of each side, we have that b ≈ 1.0274

And a  = 800/(1.0274)¹⁰  ≈ 611    And this was the initial culture size  (when t = 0, b⁰ = 1)

We can use the sine for this

sine (Θ) = opp / hyp  =  5 / 10   = 1 / 2

And this is easily evaluated, since the sine of 30° is 1/2

April's Rate ...(3/4) mi / (1/8) hr = 6 mph

Tim's Rate ....( 2/3)mi / (2/9) hr = 3 mph 

It appears that the vertex is at  (7.5, 4)....and since a parabola is symmetric about its vertex, the y value at x = 15 will have the same value as at x = 0, i.e., -3,  since x =0  and x = 15 are both 7.5 units from x = 7.5 ...so we have ...... q(15) = -3

Distance / time = Rate    ...  so we have...

120 miles / 2.5 hrs. = 48 mph

Assuming that the Sun is completely spherical and its exact center lies at (0,0)    ....and......

Depending upon the orientation of Venus's orbit ( I'll choose to have the apogee on the x axis)....we have

"a"  = .728 + .0043  = .7323

"b" =  .719 + .0043 =  .7233     so we have

x² / (a)² +  y² / (b)² = 1

x² / (.7323)² +  y² / (.7233)² = 1

x² / .53626329 +  y² / .52316289 = 1

Using the Ratio Test ( where abs = absolute value), we have

L = lim n →∞   abs [((n + 1)³ x²⁽ⁿ⁺¹⁾ ) / 5ⁿ⁺¹] * [ 5ⁿ / (n³ * x²ⁿ) ]   

L = lim n →∞   abs [[ (n+1)³ x² ] / [5 n³]]

L =(1/5) (x² ) *  lim n →∞  [ (n+1)³] /  n³ ] 

Note that  lim n →∞   [ (n+1)³] /  n³ ]  = 1      ......so we have

L = (1/5) (x² )

So if 

(1/5)(x²) < 1     the series converges  ....  and if (1/5)(x²) > 1 the series diverges

Put another way ..... if

(x²) < 5     the series converges  ....  and if (x²) > 5 the series diverges

However....the radius of convergence requires an exponent of 1 on the"x".....so we have

√(x²) < √5   →  abs (x) < √5     and   √(x²) > √5   →  abs (x) > √5 

So the radius of convergence is  R = √5

I believe this is correct......but it has been awhile since I've done this.........if anyone sees any mistakes.....feel free to provide corrections  !!!

3^(x+2)=5^x

Taking the log of both sides, we have

log3^(x+2)= log5^x    and b a property of logs, we can write

(x+2) log 3 = x log 5     simplify

x log 3   + 2 log 3  = x log 5      rearrange

2 log 3  = x log 5 - x log 3          factor on the right

2 log 3 = x (log 5 - log 3)           divide both sides by  (log 5 - log 3)

(2 log 3) / (log 5 - log 3)  = x =  about 4.3013