Question
A colony of bacteria decreases in population by 10 percent each hour. The initial bacteria culture contained 250 spores. What exponential equation can be used to find the terms in the geometric sequence?(1 point)
Responses
b(x)=250(0.99)x−1
b left parenthesis x right parenthesis equals 250 left parenthesis 0.99 right parenthesis superscript x minus 1 baseline
b(x)=250(0.10)x−1
b left parenthesis x right parenthesis equals 250 left parenthesis 0.10 right parenthesis superscript x minus 1 baseline
b(x)=250−10(x−1)
b left parenthesis x right parenthesis equals 250 minus 10 left parenthesis x minus 1 right parenthesis
b(x)=250(0.90)x−1
Responses
b(x)=250(0.99)x−1
b left parenthesis x right parenthesis equals 250 left parenthesis 0.99 right parenthesis superscript x minus 1 baseline
b(x)=250(0.10)x−1
b left parenthesis x right parenthesis equals 250 left parenthesis 0.10 right parenthesis superscript x minus 1 baseline
b(x)=250−10(x−1)
b left parenthesis x right parenthesis equals 250 minus 10 left parenthesis x minus 1 right parenthesis
b(x)=250(0.90)x−1
Answers
GPT-4o mini
To find the correct exponential equation for the population of bacteria decreasing by 10 percent each hour, we start with the initial size of the population and consider how the population changes over time.
1. The initial population is \(250\).
2. Since the population decreases by 10 percent each hour, it retains \(90\%\) of its population from the previous hour. This means that for each hour that passes, the population is multiplied by \(0.90\) (which is \(1 - 0.10\)).
The general formula for an exponential decay model is:
\[
b(x) = b_0 \cdot r^x
\]
where:
- \(b_0\) is the initial amount (in this case, \(250\)),
- \(r\) is the decay factor (which is \(0.90\) for a 10% decrease),
- \(x\) is the time in hours.
Accordingly, the equation becomes:
\[
b(x) = 250(0.90)^x
\]
However, based on the provided options, the appropriate equation that reflects the population after \(x-1\) hours is:
\[
b(x) = 250(0.90)^{x-1}
\]
But since this specific option is not explicitly listed and considering a standard representation, the correct formulation that matches the decay pattern over \(x\) hours is simply:
\[
b(x) = 250(0.90)^{x - 1}
\]
However, if we're strictly choosing from the given options, the closest option for representing the decay from the initial condition is:
\[
b(x)=250(0.90)^{x-1}
\]
But it is important to clarify that \(b(x)=250(0.90)^{x-1}\) should ideally represent the population at the end of the hour.
### Given Options:
The options provided don't match perfectly, but the one that approximates the decay theme is:
**b(x) = 250(0.90)^(x - 1)** (If listed, otherwise we'd default as follows)
**None perfectly align with standard decay modeling, hence the most suitable decay from the response provided is:**
**b(x) = 250(0.10)^(x-1)** is incorrect for decay as it does not represent \(10\%\) loss correctly.
So, the correct answer based on our deduction would resolve towards \(b(x)=250(0.90)^{x}\) if you had more functions available or the part of a sequence as in decay methods.
1. The initial population is \(250\).
2. Since the population decreases by 10 percent each hour, it retains \(90\%\) of its population from the previous hour. This means that for each hour that passes, the population is multiplied by \(0.90\) (which is \(1 - 0.10\)).
The general formula for an exponential decay model is:
\[
b(x) = b_0 \cdot r^x
\]
where:
- \(b_0\) is the initial amount (in this case, \(250\)),
- \(r\) is the decay factor (which is \(0.90\) for a 10% decrease),
- \(x\) is the time in hours.
Accordingly, the equation becomes:
\[
b(x) = 250(0.90)^x
\]
However, based on the provided options, the appropriate equation that reflects the population after \(x-1\) hours is:
\[
b(x) = 250(0.90)^{x-1}
\]
But since this specific option is not explicitly listed and considering a standard representation, the correct formulation that matches the decay pattern over \(x\) hours is simply:
\[
b(x) = 250(0.90)^{x - 1}
\]
However, if we're strictly choosing from the given options, the closest option for representing the decay from the initial condition is:
\[
b(x)=250(0.90)^{x-1}
\]
But it is important to clarify that \(b(x)=250(0.90)^{x-1}\) should ideally represent the population at the end of the hour.
### Given Options:
The options provided don't match perfectly, but the one that approximates the decay theme is:
**b(x) = 250(0.90)^(x - 1)** (If listed, otherwise we'd default as follows)
**None perfectly align with standard decay modeling, hence the most suitable decay from the response provided is:**
**b(x) = 250(0.10)^(x-1)** is incorrect for decay as it does not represent \(10\%\) loss correctly.
So, the correct answer based on our deduction would resolve towards \(b(x)=250(0.90)^{x}\) if you had more functions available or the part of a sequence as in decay methods.