A population of bacteria can be modeled by the function f(t) = 400(1/25)^t, where t is time in hours. Complete the statement about the function. The function f(t) is __ function th... A population of bacteria can be modeled by the function f(t) = 400(1/25)^t, where t is time in hours. Complete the statement about the function. The function f(t) is __ function that represents __.
Understand the Problem
The question is asking us to complete a statement about a mathematical function that models a population of bacteria over time. The function is defined with the formula provided, and we need to identify its characteristics.
Answer
The function $f(t)$ is an **exponential decay** function that represents a **decreasing population** of bacteria over time.
Answer for screen readers
The function $f(t)$ is an exponential decay function that represents a decreasing population of bacteria over time.
Steps to Solve
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Identify the function type
The function given is $f(t) = 400 \left( \frac{1}{25} \right)^t$. This is an exponential function since it includes a constant raised to the power of a variable (in this case, $t$). -
Determine the characteristics of the function
Exponential functions can model growth or decay. Since the base $\frac{1}{25}$ is less than 1, it indicates a decay in the population over time. -
Complete the statement about the function
In the context of this problem, the bacterial population decreases over time. Thus, you could say the function is decreasing. -
Confirm characteristics
Since the initial population is 400 and it decreases as $t$ increases, the function can be described as a decaying function.
The function $f(t)$ is an exponential decay function that represents a decreasing population of bacteria over time.
More Information
The function models how many bacteria remain after a certain number of hours, illustrating exponential decay due to some limiting factors in their environment.
Tips
- Confusing exponential decay with exponential growth when the base is less than one.
- Misidentifying the type of function. Remember that for a function of the form $f(t) = a \cdot b^t$, if $b < 1$, it indicates decay.
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