Find the initial amount given an exponential function.
Given the equation y = 24(0.77)^t, what is the initial amount?
To find the initial amount, I need to replace t with 0.
When I do that, I get y = 24(0.77)^0 which is y = 24(1), so y = 24
In doing other problems, that first number, where the 24 is, is always the initial amount that you start with. So I'm wondering why do I need to go through this work when the answer is just 24? Why can't I just look at the equation and say the initial amount is 24?
Is there an example where the initial value would not be what that first number would be?
Thank you.
with a little practice, which you evidently have, you can just read it from the equation, since b^0 = 1 for any nonzero b.
But, when the function is a little more complicated, you may have to plug in t=0 and evaluate a more involved expression. Read about Newton's law of cooling, or logistical growth.
In the given exponential function y = 24(0.77)^t, you are correct that the initial amount is indeed 24. In this case, the initial value is the coefficient in front of the base (0.77) raised to the power of 0. When t is 0, any number raised to the power of 0 is always 1. So, y = 24(1), and the initial amount is 24.
However, it is important to note that for most exponential functions, the initial amount might not always be evident from simply looking at the equation. In many cases, the coefficient in front of the base may not represent the initial amount directly.
For example, consider the exponential function y = 5(2)^t. In this case, although the coefficient is 5, it does not represent the initial amount. To find the initial amount, you still need to substitute t with 0: y = 5(2)^0 = 5(1) = 5. So, in this case, the initial amount is 5.
While it is true that in some cases, you can directly identify the initial amount from the equation, it is good practice to substitute the appropriate value for t to confirm the initial amount explicitly. This ensures accuracy and avoids any ambiguity.
In the given exponential function, y = 24(0.77)^t, the initial amount is indeed 24.
The reason why we go through the work of replacing t with 0 and simplifying the equation to find the initial amount is to have a clear understanding of how the exponential function behaves. By substituting t with 0, we are essentially finding the value of y when the exponent is 0, which represents the initial time.
In this particular case, when t = 0, we have y = 24(0.77)^0 = 24(1) = 24. So, the initial amount is 24.
The purpose of this process is to ensure that we are correctly interpreting the equation and understanding its implications. By substituting t with 0, we are explicitly stating that we are evaluating the expression at the initial time. This approach allows us to differentiate between the initial amount and values obtained at other points in time.
To answer your question, the initial value may not always be the number that appears first in the equation. In some cases, the equation may involve additional terms or coefficients that affect the initial amount. It's essential to carefully analyze the equation and follow the substitution process to accurately determine the initial amount.
By going through the steps of replacing t with 0, you are approaching the problem systematically and avoiding potential misconceptions. While in this specific case, it may be evident that the initial amount is 24, the substitution method ensures that you develop a general understanding and can handle more complex exponential functions.
I hope this clarifies your question. Let me know if you have any further inquiries!