1

I am a programmer, I am in a situation that requires a mathematical equation. I just hope to know the final equation that solves the problem (step by step solution is not necessary). Of course I will not use any programming jargon in describing the problem.

I developed a program for "speed reading training": You feed it with a text "story, article, ...", then it flashes these words one by one on the screen at a specified rate (speed).

If you have a text of 500 words.
You have a number of parameters to set.
Speed: number of words per minute (say 250 words/minute)

The usual expectation for the 500 words is to take 2 minutes to finish flashing word by word on the screen, it is OK.

If you have two extra correlated parameters:
Increase every & Increase amount
Increase every: Increase the speed every N milliseconds. (say 1000 milliseconds)
Increase amount: The increase amount is N words/minute (say 1 word/minute)

Now you have 500 words, and the parameters are set to;
Speed: 250 words/minute
Increase every: 1000 milliseconds
Increase amount: 1 word/minute

The program starts flashing the 500 words at a speed of 250 words/minute.
After every 1000 milliseconds the speed which is (250 words/minute) increased by one, so after the first second it will be 251 words/minute; and so on.

After finishing all the 500 words: The final speed became 349 words/minute.

Now based on the above description; if you have all these information:
Number of words (text) is 500 words
Starting speed: 250 words/minute
Increase every: unknown
Increase amount: 1 word/minute
The final speed (known in advance): 349 words/minute.

Is it possible to deduce the unknown Increase every?.

Gerry Myerson
  • 179,216
Saleh Feek
  • 111
  • It might not be a bad idea to add the programming jargon as an addendum below the question. Some people here are programmers. – Git Gud Sep 24 '13 at 06:39
  • I don't know yet, but the problem would be much simpler if you increased speed every time a word flashed rather than every $x$ milliseconds. – Jack M Sep 24 '13 at 06:42
  • It would help to say you're increasing the speed, that is to define how the increase amount and increase speed relate to the speed. If you built a formula for determining just one quantity from the others, you could use algebra to isolate the variables - if $y-x^2=0,$ $\pm\sqrt{y}=x$ and the like. – Loki Clock Sep 24 '13 at 06:57

1 Answers1

0

I haven't tested this solution, and I believe it will only be approximate due to the use of calculus (which assumes that your program is continuously accelerating rather than in discrete bursts once a second). I know you didn't ask for a step by step solution, but I'm posting it anyway so others (and yourself) can criticize it.

If I understood your problem correctly, you flash words at a certain rate. Once per, say, second (the amount of time is actually the variable we want to solve for), you increase this rate by a fixed number of words per minute. You have a fixed amount of words to get through, and you want to hit a certain top speed by the time you get through all those words. You want to know how often to increase the speed in order to achieve this.

Letting $v_i$ be your initial speed (in words/second), $a$ be the amount by which you additively increase this speed, $x$ be the delay in between successive increases in speed (in seconds), and $t$ the amount of time in seconds since program start, the number of words flashed by time $t$ is (approximately) an antiderivative:

$$\int v_i+a\frac{t}{x}dt$$

And thus, given that at $t=0$ you've flashed zero words, we have that the number of words flashed by time $t$ is:

$$v_it+a\frac{t^2}{2x}$$

Setting this equal to the total number of words $W$ and solving for $t$, we obtain a value for $t_f$, the amount of time the program takes to run. Reinsert that into the expression for speed to obtain the final speed, and set that equal to $v_f$, the desired final speed.

$$v_i+a\frac{t_f}{x}=v_f$$

Solve for $x$.

The solution is, I believe, $$x=\frac{2aW}{v_i^2-v_f^2}$$

Jack M
  • 27,819
  • 7
  • 63
  • 129