Thread: Creating exponential equations

1. Creating exponential equations

Hello,

I have the two equations:

h(x) = 3(3)^x
f(x) = 100(2)^(x/32)

h(x) is the number of people told during the hour, x, and x = 0, corresponds to the hour which the first 3 people heard the rumour and started telling others.

f(x) is the total number of bacteria within a culture which doubles every 0.32 hours, and x is the number of hours. x = 0 corresponds to the initial amount of bacteria (which is 100).

This is the dilemma that I am having. Both equations are very similar, and yet, h(x) represents the total additional amount of people told based on the number of hours since the beginning and f(x) represents the total amount of bacteria based on the number of hours since the beginning.

Is there any fundamental reason why one represents the total amount versus another only represents the total additional amount? I'm having a hard time wrapping my head around this. I did not create these equations. I feel like it is important for me to understand the difference in order to be able in the future to derive equations which represent the context of the situation correctly.

2. Re: Creating exponential equations

If I understand this, you are wondering why, when one s the "additional amount" and the other is the "total amount", the formulas are so similar. That is a property of the exponential function. For any base, a, $a^{x+y}= a^x a^y$. Here "y" is the "additional amount" and "x+ y" is the total amount. The second is just a constant, $a^x$, times the first.

3. Re: Creating exponential equations

Yes, I'm saying that h(x) is a formula apparently that calculates the additional amount, while f(x) calculates the total amount.

I understand that there are two numbers within the equation of f(x) that have the same base and hence why you can apply exponent properties to them, but I don't see how this answers my question.

I am facing a word problem in the textbook that says f(x) measures the additional amount of the variable x. Whereas h(x) measures the total amount over the variable of x. I'm confused, since h(x) is so similar to f(x), why doesn't h(x) also measure the additional amount of x? Or why doesn't f(x) measure the total amount over the variable of x? It is the specific wording that confuses me. The context of the word problems are so different, as in one is measuring the additional amount while the other is measuring the total, and yet there equations are very similar. I'm surprised how this could even be correct.

Do you understand what I mean?

4. Re: Creating exponential equations

I'm not sure I do understand what you mean.

5. Re: Creating exponential equations

Originally Posted by otownsend
Hello,

I have the two equations:

h(x) = 3(3)^x
f(x) = 100(2)^(x/32)

h(x) is the number of people told during the hour, x, and x = 0, corresponds to the hour which the first 3 people heard the rumour and started telling others.

f(x) is the total number of bacteria within a culture which doubles every 0.32 hours, and x is the number of hours. x = 0 corresponds to the initial amount of bacteria (which is 100).

This is the dilemma that I am having. Both equations are very similar, and yet, h(x) represents the total additional amount of people told based on the number of hours since the beginning and f(x) represents the total amount of bacteria based on the number of hours since the beginning.

Is there any fundamental reason why one represents the total amount versus another only represents the total additional amount? I'm having a hard time wrapping my head around this. I did not create these equations. I feel like it is important for me to understand the difference in order to be able in the future to derive equations which represent the context of the situation correctly.
There is absolutely no mathematical relationship between h(x) and f(x). h(x) is a mathematical expression for passing rumors and f(x) is a mathematical expression for bacteria growth. In each expression, the author of the expression merely chose x as his variable, It's just merely coincidence that h(x) looks similar to f(x). The author of f(x) could have just as correctly chosen b (for bacteria) as his variable and f(x) would have then been written as f(b) = 100(2)^(b/32).

Steve

6. Re: Creating exponential equations

Ok I see what you mean now...thank you.

7. Re: Creating exponential equations

I had assumed that this was a general question with two completely separate example.