CPhill

LOL!!!!

The "formulas" for the first two are similar.......just the times compounded per year are different

A = P (1 + r / n)^(n *t)     where P is the principal, r is the interest rate, n is the number of compoundings per year and t is the number of years

A)  Yearly

5000(1 + .085/ 1)^(1 * 6 )  ≈  $8157.34

B) Monthly

5000(1 +.085/12)^(12* 6)   ≈ $ 8311.50

C)  Continuously ......   use  A =    Pe^(rt)      

5000e^(.085 * 6)  ≈  $8326.46

We are using   log(a * b)   = log a+ log b    ......so.......

log (4m^3n^7)  =  

log 4 + log m^3  + log n^7

Now we can use    log a^b    =  b log a

log 4 + 3 log m  + 7 log n

Well....it could be lots of things.....but....let's try this one.........

Let's assume that this is the bottom half of a sideways parabola opening to the right

Let's call the vertex  (h,k)  =  (5 ,6)     and we know that point  (9, -2)  is on the graph.......so........we have this form :

(x - 5) = a(y - 6)^2      now......substitute 9 for x and -2 for y   and solve for a......and we have

(9 -5)  = a(-2 - 6)^2

4  = a(64)   divide both sides by 64

a = 4/64   = 1/16

So........our full equation is

(x - 5)  = (1/16)(y - 6)^2     but we only want the bottom part of the graph....so....    

Multiply both sides by 16

16(x - 5)  = (y - 6)^2         take the pos/neg square roots of both sides

± 4 √ (x - 5)  = y  - 6

Since we want the bottom part, we will take the negative root

- 4 √ (x - 5)  = y  - 6     add 6 to both sides

-4 √ (x - 5) + 6  = y    and we can write this as :

y = -4 √ (x - 5) + 6 

Here's the graph : https://www.desmos.com/calculator/tpovquifna

I got the same result as Alan did.....as a check......

y - p^2x   .......     let x = 5p - 5/p  .....    and let y = -x      ....   and we have....   

[5/p - 5p] - p^2 [ 5p - 5/p]  =

5/p - 5p - 5p^3 + 5p  =

5/p - 5p^3       and this  =  the right side

So.......the solution should be :

(x , y)   =   ( 5p - 5/p ,  5/p - 5p )

(4x^3+2x^2-5)/2x-3    let's write this as

[4x^3 + 2x^2 + 0x  - 5 ]  /  [2x - 3]        now.....let's perform polynomial division

                2x^2   + 4x  + 6

2x - 3   [  4x^3 + 2x^2 + 0x  - 5 ]

               4x^3 -  6x^2

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

                           8x^2  + 0x

                           8x^2  - 12x

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

                                       12x - 5

                                       12x - 18

                                        -----------

                                                13

So our answer is :   2x^2 + 4x + 6  +  13 / [2x - 3]

That's pretty crafty, heureka.........!!!........thanks for showing me how you derived it.....

5x-8=13     add 8 to both sides

5x = 21      divide both sides by 5

x = 21 / 5

Heureka probably knows how to do this better with vectors.......and maybe even come up with more values for y than I did......but, here's my attempt  :

The first value for y is pretty easily derived.......let the base be defined the two points (-5,1) and (-4,y).........if we let y = 1.........the length of the base = 1 and the height of the oblique triangle [ with  (0,5) as the "apex" point ] will be 4.....so......the area of this triangle =  (1/2)(b)(h)  = (1/2)(1)(4)  = 2 sq units

To get a second point....let the base be defined by the two points  (-5,1) and (0,5).....and the length of this base = sqrt (41)

And the equation of the line going through these two points is

y   = (4/5)x + 5   which we can write as    (4/5)x - y + 5  = 0

Using the area formula for a triangle, we can calculate the needed height  of our second triangle

2 = sqrt(41)/2 *h    →  h  = 4/sqrt(41)

So........we need to find a point with an x coordinate of -4 that is 4/sqrt(41)   units from the line

y = (4/5)x + 5

Using the equation for the perpendicular distance from a point to a given line, we have

abs [ (4/5)(-4)  -1y + 5 ] / ( sqrt [ 16/25 + 1 ] )  =  4/sqrt(41)

abs [ -16/5 -1y + 5 ] / sqrt (41)/ 5  = 4/sqrt(41)

abs [ 9/5 - y] * 5  = 4

abs [9/5 -y] = 4/5

And we have two equations here

9/5 - y  = 4/5                 or                  9/5 - y   = - 4/5

y = 9/5  - 4/5  = 1                                 y =  9/5 + 4/5  =  13/5   = 2.6

Notice that this gave us not only a second point, but also my original "guess" as well

So.......the two values of y are :     1, 2.6

Here's an  image of both triangles that we found......the first is triangle ABC   and the second is triangle ABD

In one hour, Nick does 1/12 of the job, Adam does 1/6 of the job and Mike does 1/8 of the job

So.......  1/12 + 1/6 + 1/8  =  3/8 of the job is done after one hr  and 6/8 = 3/4 of the job is done after two hours.....so  only 1/4 remains = 6/24

When Adam leaves......1/12 + 1/8  = 5/24 of the job gets done in one hr.....so....it will take the two other guys working together another  [ 6/24] / [5/24]  = 6/5 of an hour to finish = 72 more minutes

So.....the whole job is completed in 3 hrs and 12 miniutes.........therefore......Nick needn't worry

Here you go :  https://www.desmos.com/calculator/eq4xqzaeww