CPhill

https://web2.0calc.com/questions/geometry_93750

Simplify as 

3x^2 - 32x  +  c

We will have a square of a binomial when the  discriminant  = 0

32^2  -  4(3)c  =  0

1024  =  12c

c = 1024/ 12 =  256/3

Law of Cosines

PR^2  = PQ^2 +QR^2   -2(PQ * QR)cos ( PQR)

23^2  =18^2 + 25^2  - 2(18 * 25) cos (PQR)

[23^2 - 18^2 - 25^2 ] / [ -2 * 18 * 25 ] =  cos (PQR)

7/15 = cos (PQR)

Law of Cosines again

PM^2  = PQ^2 + (QR/2)^2 - 2(PQ * QR/2) cos (PQR)

PM^2  = 18^2 +(12.5)^2  - 2 (18 * 12.5)(7/15)

PM^2 = 270.25

PM = sqrt (270.25)  ≈  16.44

Alan has answered that question here: https://web2.0calc.com/questions/please-help_63707

Equations of the three medians

y  = x                                                       

y = (-2)x + 1      ( 1)

y = (-1/2)x + 1/2           (2)

x intersection of (1) , (2)

-2x + 1  = -(1/2)x + 1/2

(3/2)x = 1/2

x = 1/3

y = (-2)(1/3) + 1 =   -2/3 + 1  =  1/3

They pass through rhe point  (1/3 , 1/3)

Your team  can never win 60% of its games  because you only have a percentage of 1/12 ≈  8.33%

And if you  only win 1 game  per week  and lose 2, you can never raise your winning percentage to 60%

\((3-\sqrt{5})(4+\sqrt{5})(1+6\sqrt{5})\)

Product of first two

( 12 -sqrt 5) - 5)  =  (7 - sqrt (5) )

(7  -sqrt 5) ( 1+ 6sqrt 5)   =   7  + 41sqrt 5   - 30   =  -23 + 41sqrt (5)

Let the load that your truck  can  carry   =  x

Let the load that each of  your neighbors trucks can carry =  y

We have

x  -  y   ≥  500

x ≤ (3/8) (4y)      →  x ≤ (3/2)y

See the graph here :

Max your tuck can carry  =1500 lbs

\( 9^x - 2 \cdot 3^x + 1 \)

Take the derivative and  set to 0

9^x ( ln 9)  - 2 * 3^x (ln 3)  = 0

9^x (ln 9)  - ( 3^x) ( 2ln3)  = 0

9^x * (ln 9)  - (ln 3^2) * 3^x  = 0

(ln 9)  ( 9^x - 3^x)   = 0

9^x  - 3^x   = 0

9^x   = 3^x

(3^2)^(x)  = 3^(x)        take the log of both sides

log (3)^(2x) = log(3)^x

(2x) log 3  =  x log 3

(log 3) (2x - x)  = 0

2x - x  = 0

x = 0

The min is

9^(0)  - 2* 3^(0)  + 1 = 

1  -  2  + 1 =    0

Points on the graph =  (0,1)  and (1,0)

Slope of line =  [ 0 -1 ] / [ 1 - 0 ] = -1 / 1   = -1

Equation of line      y =  -x + 1     →   x + y =  1

Equation  of inequality is either

x + y   ≥ 1

or

x + y ≤  1

(1,1)  is in the solution area  and  makes  the first inequality true