Math Problem Statement
Solution
The problem provides an exponential function to model the growth of bacteria on a kitchen counter. Here's the given function:
Where:
- represents the number of bacteria at time (in minutes),
- is the time in minutes since the initial count of 10 bacteria,
- is the base of the natural logarithm.
The problem asks how many bacteria will be on the counter 2.5 hours later. Let's break it down step by step:
Step 1: Convert Hours to Minutes
Since the function uses time in minutes, we need to convert 2.5 hours to minutes:
Step 2: Plug Time into the Exponential Model
Now, substitute into the function:
Step 3: Calculate the Value of the Exponent
Step 4: Final Calculation
Now multiply by 10:
Step 5: Round to the Nearest Whole Number
Thus, after 2.5 hours, the predicted number of bacteria is approximately 325,575.
Would you like more details or further clarifications?
Related Questions:
- What is the significance of the base in exponential growth functions?
- How would the result change if the doubling time were different, say 20 minutes instead of 10?
- How can the constant 0.06931 be derived from the doubling time of 10 minutes?
- How would the equation change if we started with a different initial amount of bacteria?
- Can we apply this type of model to other real-life scenarios, such as population growth or radioactive decay?
Tip:
In exponential growth problems, small changes in time or the growth rate can lead to significantly different results due to the nature of exponential functions.
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Math Problem Analysis
Mathematical Concepts
Exponential growth
Natural logarithms
Formulas
A(t) = 10e^0.06931t
Theorems
Exponential growth model
Suitable Grade Level
High school (Grades 10-12)
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