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# {CS Homework} Modeling 1 Class

### Technology

y = 0.75x
This is the equation of a line with slope m=0.75.
The corresponding inputs and outputs for this line (16,12), (12,9), (9,6.75), ...

How do we get from one point to the next? In particular, how can the point (a(1),a(2)) = (12,9) be used to find the next point (a(2),a(3)) = (9,6,75)?

Is there a smart gamer who can explain why they're asking me this?

Starfinder Charter Superscriber

Who are "they" that are asking you? I can't attempt to explain "why" without knowing "who".

Pathfinder Roleplaying Game Subscriber

Slope is a function of the change in coordinate b for every unit of coordinate a. In geometry it's commonly called rise over run (y /x).

In your example, slope of 0.75 means that for every 1 unit of a, b changes by 0.75. From 16,12 to 12,9, coordinate a has a change of -4 units. To verify the slope, we can multiply the slope by the change. -4 * 0.75 = -3. | 12 - 3 = 9.

And again for the second pair, change of -3 * slope of 0.75 = -2.25. 9 - 2.25 = 6.75.

Distant Scholar wrote:
Who are "they" that are asking you? I can't attempt to explain "why" without knowing "who".

I'm taking this man's modeling class. http://www.ibspan.waw.pl/~kacprzyk/glowna/en

Yesterday he said this equation pertains to an ancient type of analysis named cobb web analysis. He handed out a short tome on the subject and, get this, it's partially written in sanskrit.

I'm in a computer science class so I was a bit surprised, and last night was hunting around for some insight. A bit of background, he said hindu scribes from around 300 AD first discovered this type of analysis and used it while mixing dye for producing colored clothe. And now here I am learning about it in Modeling 1 class.

Those ancient clothe makers didn't play around:

1. Suppose a bowl contains five cups of water. Initially, add 20 ml of die to the bowl. Then every 4 hours, replace one cup of water with a cup of water containing 10 ml of dye.
- a. Develop a dynamical system for a(n), the amount of dye in the bowl after n four-hour time periods.
- b. How much dye will be in the bowl at the end of 12 hours?
- c. What is the equilibrium value for this dynamical system?
- d. Sketch a cobweb diagram for this problem.

And now the fun began:

Starfinder Charter Superscriber

I imagine you're being asked this question as an exercise in recursion / dynamical systems / Markov chains / state machines. Pick your favorite; they're all related to each other.

A lot of number-theoretical concepts have been around for a very, very long time, and are still quite useful.

Now we are doing starter walkthrough code to numerically walk the "cobb web" and find equilibriums. At least that's what I was told would happen.

Each four hours, you remove 1/7th of the whole, which starts at 20 ml of dye in total 70 ml of fluid, replacing it by 10 ml of dye. Thus, the amount of dye in the fluid a(n+1)=6/7*a(n)+10. a(0)=20. If I didn't miss anything.

Here is a much older version of my text book. I offer it as a way to play along at home.

I got:

a.
a(n) = (4/5)*a(n-1) + 10
a(0) = 20

b.
a0 = 20;
a1 = (4/5)*a0+10 = 26
a2 = (4/5)*a1+10 = 30.8
a3 = (4/5)*a2+10 = 34.64

At 12hrs, dye amount = 34.64mL. Because n steps off 4-hour segments.

Oh, we're using javascript in this class. This doesn't bother me, and .js is so available my opinion is positive.

repl.it/languages/javascript

c.
a = 50

d.
(I graphed it by hand.)

And then the band started playing. So in response, teacherMan asked this question next.

2. Consider the dynamical system

a(n+1) = 2a(n) - 0.25a²(n) - 0.75

The two equilibrium values are a = 1, which is unstable, and a = 3, which is stable. Sketch a cobweb for this equation and use it to determine the maximum interval containing a = 3 such that if a(0) is in this interval, then lim_{k \to infinity} a(k) = 3.

I'm using colored pencils because they're nice.

I'm back again for a few hours. It's near 1am and I study best in the dark.
Now the sun is asleep, and malevolent entities forage in the night hunting for knowledge. Or perhaps they are more pernicious and seek to prey on humans lost in the darkness. To find one in its grasp, and eat the still warm brain.

"Run", she screams.

Black legs scramble across the ground. There are too many legs for the creature to be normal. The guttural voice sounding more like a coughing bull proves it not to be.

"D... Die," it booms out.

James goes down underneath it. He probably never heard her scream. Now her hands are shaking with fear and adrenaline as she tries to fire up a 'repl.it javascript' editor.

The light of the editor showed life in her face.

"Code", she thought. "Write code."

Only code will defeat this beast from the dark.

note: this is called procrastination ...

Tonight's endevour... make (a(n-1),a(n)) |--> (a(n),a(n)) work. I think work is the correct word.

n starts at 0.
So, compute a(0) quickly followed by a(1).
Now we have a(1), so compute a(2).

I don't see why this is so difficult. That means I'm missing something.

Great.