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Sagot :
To determine an equivalent form of the equation [tex]\(\rho(t) = 20,000 \cdot 2^{\frac{t-7}{10}}\)[/tex] for the bacteria population in the second sample, we need to simplify and verify the given options.
First, let's rewrite the given equation in a more convenient form:
[tex]\[ \rho(t) = 20,000 \cdot 2^{\frac{t-7}{10}} \][/tex]
We can simplify this expression for clarity. Note that the exponent [tex]\(\frac{t-7}{10}\)[/tex] can be broken down using the properties of exponents:
[tex]\[ \rho(t) = 20,000 \cdot 2^{\frac{t}{10} - \frac{7}{10}} \][/tex]
Using the property [tex]\(a^{b-c} = \frac{a^b}{a^c}\)[/tex], we rewrite this as:
[tex]\[ \rho(t) = 20,000 \cdot \frac{2^{\frac{t}{10}}}{2^{\frac{7}{10}}} \][/tex]
Therefore, we simplify the equation:
[tex]\[ \rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}} \][/tex]
Now, we compare this simplified form with the given options to see which one is equivalent. The options to consider are:
1. [tex]\(p(t) = \frac{20,000 \cdot 2^{t-7}}{2^{10}}\)[/tex]
2. [tex]\(p(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7}\)[/tex]
3. [tex]\(\rho(t) = \frac{20,000 \cdot \sqrt[7]{2^t}}{2^{10}}\)[/tex]
Let's evaluate these options individually:
1. Option 1: [tex]\(\frac{20,000 \cdot 2^{t-7}}{2^{10}}\)[/tex]
Simplifying the exponent in the numerator:
[tex]\[ 2^{t-7} = 2^t \cdot 2^{-7} = \frac{2^t}{2^7} \][/tex]
So:
[tex]\[ \frac{20,000 \cdot 2^{t-7}}{2^{10}} = \frac{20,000 \cdot \frac{2^t}{2^7}}{2^{10}} = \frac{20,000 \cdot 2^t}{2^{7+10}} = \frac{20,000 \cdot 2^t}{2^{17}} \][/tex]
This is not equivalent to [tex]\(\rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}}\)[/tex].
2. Option 2: [tex]\(\frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7}\)[/tex]
Simplifying this directly:
[tex]\[ \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7} = \frac{20,000 \cdot 2^{\frac{t}{10}}}{128} \][/tex]
This is also not equivalent to [tex]\(\rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}}\)[/tex].
3. Option 3: [tex]\(\frac{20,000 \cdot \sqrt[7]{2^t}}{2^{10}}\)[/tex]
Writing the square root in exponential form:
[tex]\[ \sqrt[7]{2^t} = (2^t)^{\frac{1}{7}} = 2^{\frac{t}{7}} \][/tex]
So:
[tex]\[ \rho(t) = \frac{20,000 \cdot 2^{\frac{t}{7}}}{2^{10}} \][/tex]
This is certainly not equivalent to [tex]\(\rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}}\)[/tex].
Given the analysis:
1. Option 1 is not equivalent to the simplified form.
2. Option 2 is not equivalent to the simplified form.
3. Option 3 is also not equivalent to the simplified form.
Thus, none of the given options [tex]\(( \rho(t) = \frac{20,000 \cdot 2^{t-7}}{2^{10}}, \rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7}, \text{or} \rho(t) = \frac{20,000 \cdot \sqrt[7]{2^t}}{2^{10}} )\)[/tex] are equivalent to the original expression [tex]\(\rho(t) = 20,000 \cdot 2^{\frac{t-7}{10}}\)[/tex].
First, let's rewrite the given equation in a more convenient form:
[tex]\[ \rho(t) = 20,000 \cdot 2^{\frac{t-7}{10}} \][/tex]
We can simplify this expression for clarity. Note that the exponent [tex]\(\frac{t-7}{10}\)[/tex] can be broken down using the properties of exponents:
[tex]\[ \rho(t) = 20,000 \cdot 2^{\frac{t}{10} - \frac{7}{10}} \][/tex]
Using the property [tex]\(a^{b-c} = \frac{a^b}{a^c}\)[/tex], we rewrite this as:
[tex]\[ \rho(t) = 20,000 \cdot \frac{2^{\frac{t}{10}}}{2^{\frac{7}{10}}} \][/tex]
Therefore, we simplify the equation:
[tex]\[ \rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}} \][/tex]
Now, we compare this simplified form with the given options to see which one is equivalent. The options to consider are:
1. [tex]\(p(t) = \frac{20,000 \cdot 2^{t-7}}{2^{10}}\)[/tex]
2. [tex]\(p(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7}\)[/tex]
3. [tex]\(\rho(t) = \frac{20,000 \cdot \sqrt[7]{2^t}}{2^{10}}\)[/tex]
Let's evaluate these options individually:
1. Option 1: [tex]\(\frac{20,000 \cdot 2^{t-7}}{2^{10}}\)[/tex]
Simplifying the exponent in the numerator:
[tex]\[ 2^{t-7} = 2^t \cdot 2^{-7} = \frac{2^t}{2^7} \][/tex]
So:
[tex]\[ \frac{20,000 \cdot 2^{t-7}}{2^{10}} = \frac{20,000 \cdot \frac{2^t}{2^7}}{2^{10}} = \frac{20,000 \cdot 2^t}{2^{7+10}} = \frac{20,000 \cdot 2^t}{2^{17}} \][/tex]
This is not equivalent to [tex]\(\rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}}\)[/tex].
2. Option 2: [tex]\(\frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7}\)[/tex]
Simplifying this directly:
[tex]\[ \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7} = \frac{20,000 \cdot 2^{\frac{t}{10}}}{128} \][/tex]
This is also not equivalent to [tex]\(\rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}}\)[/tex].
3. Option 3: [tex]\(\frac{20,000 \cdot \sqrt[7]{2^t}}{2^{10}}\)[/tex]
Writing the square root in exponential form:
[tex]\[ \sqrt[7]{2^t} = (2^t)^{\frac{1}{7}} = 2^{\frac{t}{7}} \][/tex]
So:
[tex]\[ \rho(t) = \frac{20,000 \cdot 2^{\frac{t}{7}}}{2^{10}} \][/tex]
This is certainly not equivalent to [tex]\(\rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^{\frac{7}{10}}}\)[/tex].
Given the analysis:
1. Option 1 is not equivalent to the simplified form.
2. Option 2 is not equivalent to the simplified form.
3. Option 3 is also not equivalent to the simplified form.
Thus, none of the given options [tex]\(( \rho(t) = \frac{20,000 \cdot 2^{t-7}}{2^{10}}, \rho(t) = \frac{20,000 \cdot 2^{\frac{t}{10}}}{2^7}, \text{or} \rho(t) = \frac{20,000 \cdot \sqrt[7]{2^t}}{2^{10}} )\)[/tex] are equivalent to the original expression [tex]\(\rho(t) = 20,000 \cdot 2^{\frac{t-7}{10}}\)[/tex].
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