Alan

There are several ways to find the approximate value of the square root of a number.  One of the simplest is to use Newton's method.  If you want to find the square root of N, start with an initial guess x₀, then iterate using the following:  

x_n+1 = (x_n + N/x_n)/2    where n is the iteration number. (i.e.   n = 0, 1, 2, 3, ... etc.)

Continue iterating until the result is as accurate as desired.

c)  The tangent line parallel to y = -12x will have slope -12, so will be of the form y = -12x + k, where k is a constant. 

   We must have  -2x^-3 + 4  = -12 for the slopes to match, which gives us the value of x at which this tangent line touches the curve.  Plug this value into y = x^-2 + 4x to find the corresponding value of y.  Plug these values of x and y into  y = -12x + k and rearrange to find k.

-2x^-3 + 4  = -12 → x^-3 = 8  → x = 1/2    

The corresponding value of y is (1/2)^-2 + 4(1/2) → 4 + 2 → 6

So  6 = -12*(1/2) + k   →   6 = -6 + k  →  k = 12

Hence   y = -12x + 12 is the equation of the line tangent to the curve and parallel to y = -12x,

See my reply here: https://web2.0calc.com/questions/math-question_125#r5 

I get k = 1.33 using the data provided in the question.  In order to get k = 1.11 I have to assume that Bolt runs 100 m in 9.85 s and Wesno runs 100 bars in 11.85 ticks!

This was answered here: https://web2.0calc.com/questions/a-curve-has-equation-y-1-x-2-4x#r1 

Here is the how to find your circumcenter:

Here's my take on this:

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"Annual consumption of bottled water in the United States is given in the table ... etc"

The following should help. Note that I've used a slightly different notation, and I've not shown how I obtained the best-fit values):

(I should add that extrapolating an empirical fit this far from the experimental data is a silly thing to do in general!!)

"Let C be a point not on line AE and D a point on line AE such that CD is perpendicular to AE Meanwhile, B is a point on line CE such that AB is perpendicular to CE.If AB=4, CD=8 and AE=5 then what is the length of CE"

Conservation of momentum:   m₁v₁ + m₂v₂ = 0

Let subscript 1 represent the shot mass, and subscript 2 the gun. m is mass, v is velocity

100*400 + 1000*v₂ = 0  so  v₂ =   -40 m/s      (I assume you meant m/s for velocity, not ms !!)

 Note that the negative sign indicates the gun travels in the opposite direction to the shot.

Assuming constant acceleration (deceleration here) of the gun we can use the constant linear acceleration equation 

v² = u² + 2as  to find the acceleration, a:

0 = (-40)² + 2a*(-0.12)   so a = 1600/0.24  m/s²   (Note: I'm assuming direction is positive in the direction of the shot)

Newton's second law says force = mass*acceleration, so F = 1000*1600/0.24  N