Question
Hours (x) 0 1 2 3 4 5
Number of Bacteria (y) 23 53 123 282 650 1,493
The table shows the growth of a type of bacteria. Rounded to the nearest tenth, which equation best models this data?
Responses
A y = 266x - 227y = 266 x - 227
B y = (23)
y = (23) ( 2.3 ) x
C y = (2.3)
y = (2.3) ( 23 ) x
D y = 93
- 2.037x + 86
Number of Bacteria (y) 23 53 123 282 650 1,493
The table shows the growth of a type of bacteria. Rounded to the nearest tenth, which equation best models this data?
Responses
A y = 266x - 227y = 266 x - 227
B y = (23)
y = (23) ( 2.3 ) x
C y = (2.3)
y = (2.3) ( 23 ) x
D y = 93
- 2.037x + 86
Answers
Answered by
GPT-4o mini
To determine the best equation that models the growth of the bacteria based on the given data, let's analyze the data points and the responses provided.
The values in the table show an exponential growth pattern (the number of bacteria appears to increase rapidly). The data points can be plotted or inspected to look for an exponential growth pattern.
Let's analyze the responses one by one:
A. **y = 266x - 227**
- This is a linear equation. Given the nature of the data, a linear equation would not be a good fit since we have exponential growth.
B. **y = (23)(2.3)^x**
- This is an exponential equation. Starting with 23 and increasing by a factor of 2.3 every hour could fit the growth of bacteria. We would need to check the values.
C. **y = (2.3)(23)^x**
- This is also an exponential equation but suggests multiplying by a constant for each x which does not fit the growth model as effectively.
D. **y = 93 - 2.037x + 86**
- This one looks like a combination of linear terms and is quite complicated. This also doesn't appear to fit the exponential growth pattern.
To verify which option fits the best, let's calculate some values for option B:
Using \(y = 23(2.3)^x\):
- For \(x = 0\): \(y = 23(2.3)^0 = 23\)
- For \(x = 1\): \(y = 23(2.3)^1 = 52.9\) (rounds to 53)
- For \(x = 2\): \(y = 23(2.3)^2 \approx 121.69\) (rounds to 123)
- For \(x = 3\): \(y = 23(2.3)^3 \approx 279.2\) (rounds to 282)
- For \(x = 4\): \(y = 23(2.3)^4 \approx 646.6\) (rounds to 650)
- For \(x = 5\): \(y = 23(2.3)^5 \approx 1493.7\) (rounds to 1493)
Since all values closely match the data, option **B** \(y = (23)(2.3)^x\) is indeed the best model to represent the growth of the bacteria.
Therefore, the correct response is:
**B. y = (23)(2.3)^x**
The values in the table show an exponential growth pattern (the number of bacteria appears to increase rapidly). The data points can be plotted or inspected to look for an exponential growth pattern.
Let's analyze the responses one by one:
A. **y = 266x - 227**
- This is a linear equation. Given the nature of the data, a linear equation would not be a good fit since we have exponential growth.
B. **y = (23)(2.3)^x**
- This is an exponential equation. Starting with 23 and increasing by a factor of 2.3 every hour could fit the growth of bacteria. We would need to check the values.
C. **y = (2.3)(23)^x**
- This is also an exponential equation but suggests multiplying by a constant for each x which does not fit the growth model as effectively.
D. **y = 93 - 2.037x + 86**
- This one looks like a combination of linear terms and is quite complicated. This also doesn't appear to fit the exponential growth pattern.
To verify which option fits the best, let's calculate some values for option B:
Using \(y = 23(2.3)^x\):
- For \(x = 0\): \(y = 23(2.3)^0 = 23\)
- For \(x = 1\): \(y = 23(2.3)^1 = 52.9\) (rounds to 53)
- For \(x = 2\): \(y = 23(2.3)^2 \approx 121.69\) (rounds to 123)
- For \(x = 3\): \(y = 23(2.3)^3 \approx 279.2\) (rounds to 282)
- For \(x = 4\): \(y = 23(2.3)^4 \approx 646.6\) (rounds to 650)
- For \(x = 5\): \(y = 23(2.3)^5 \approx 1493.7\) (rounds to 1493)
Since all values closely match the data, option **B** \(y = (23)(2.3)^x\) is indeed the best model to represent the growth of the bacteria.
Therefore, the correct response is:
**B. y = (23)(2.3)^x**
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