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To determine the values of [tex]\( A \)[/tex], [tex]\( b \)[/tex], [tex]\( x \)[/tex], and [tex]\( y \)[/tex] in the exponential growth equation for the bacterial cells, follow these steps:
1. Identifying the General Form of the Exponential Growth Equation:
An exponential growth equation typically has the form:
[tex]\[ y = A \cdot b^x \][/tex]
where
- [tex]\( y \)[/tex] is the number of bacterial cells,
- [tex]\( A \)[/tex] is the initial amount of cells,
- [tex]\( b \)[/tex] is the base representing the growth factor,
- [tex]\( x \)[/tex] is the time in hours.
2. Determine the Initial Amount of Cells ([tex]\(A\)[/tex]):
From the given data, at time [tex]\( x = 0 \)[/tex] hours, the number of cells [tex]\( y \)[/tex] is 100. Therefore, the initial amount ([tex]\( A \)[/tex]) is:
[tex]\[ A = 100 \][/tex]
3. Calculate the Growth Factor ([tex]\(b\)[/tex]):
We can use the data from [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex] to find [tex]\( b \)[/tex]:
- At [tex]\( x = 0 \)[/tex] hours, [tex]\( y = 100 \)[/tex]
- At [tex]\( x = 1 \)[/tex] hour, [tex]\( y = 200 \)[/tex]
The equation at [tex]\( x = 1 \)[/tex] hour is:
[tex]\[ 200 = 100 \cdot b \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ b = \frac{200}{100} = 2 \][/tex]
4. Identify the Exponential Equation:
Substitute [tex]\( A = 100 \)[/tex] and [tex]\( b = 2 \)[/tex] into the general form. The exponential equation for the number of cells is:
[tex]\[ y = 100 \cdot 2^x \][/tex]
5. Find the Number of Hours ([tex]\(h\)[/tex]) Needed for 6400 Cells:
When [tex]\( y = 6400 \)[/tex], we need to solve for [tex]\( x \)[/tex] (which we denote as [tex]\( h \)[/tex]):
[tex]\[ 6400 = 100 \cdot 2^h \][/tex]
First, divide both sides by 100:
[tex]\[ 64 = 2^h \][/tex]
Using properties of exponents, we know:
[tex]\[ 2^6 = 64 \][/tex]
Thus,
[tex]\[ h = 6 \][/tex]
6. Final Exponential Equation:
Based on the calculated values, the exponential equation for the growth of the bacterial cells, accounting for the given values, is:
[tex]\[ y = 100 \cdot 2^x \][/tex]
Specifically, for [tex]\( h = 6 \)[/tex]:
[tex]\[ 6400 = 100 \cdot 2^6 \][/tex]
Therefore, this detailed, step-by-step solution shows that the values for [tex]\( A, b, h \)[/tex], and [tex]\( y \)[/tex] are 100, 2, 6, and 6400, respectively, in the context of exponential growth of bacterial cells.
1. Identifying the General Form of the Exponential Growth Equation:
An exponential growth equation typically has the form:
[tex]\[ y = A \cdot b^x \][/tex]
where
- [tex]\( y \)[/tex] is the number of bacterial cells,
- [tex]\( A \)[/tex] is the initial amount of cells,
- [tex]\( b \)[/tex] is the base representing the growth factor,
- [tex]\( x \)[/tex] is the time in hours.
2. Determine the Initial Amount of Cells ([tex]\(A\)[/tex]):
From the given data, at time [tex]\( x = 0 \)[/tex] hours, the number of cells [tex]\( y \)[/tex] is 100. Therefore, the initial amount ([tex]\( A \)[/tex]) is:
[tex]\[ A = 100 \][/tex]
3. Calculate the Growth Factor ([tex]\(b\)[/tex]):
We can use the data from [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex] to find [tex]\( b \)[/tex]:
- At [tex]\( x = 0 \)[/tex] hours, [tex]\( y = 100 \)[/tex]
- At [tex]\( x = 1 \)[/tex] hour, [tex]\( y = 200 \)[/tex]
The equation at [tex]\( x = 1 \)[/tex] hour is:
[tex]\[ 200 = 100 \cdot b \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ b = \frac{200}{100} = 2 \][/tex]
4. Identify the Exponential Equation:
Substitute [tex]\( A = 100 \)[/tex] and [tex]\( b = 2 \)[/tex] into the general form. The exponential equation for the number of cells is:
[tex]\[ y = 100 \cdot 2^x \][/tex]
5. Find the Number of Hours ([tex]\(h\)[/tex]) Needed for 6400 Cells:
When [tex]\( y = 6400 \)[/tex], we need to solve for [tex]\( x \)[/tex] (which we denote as [tex]\( h \)[/tex]):
[tex]\[ 6400 = 100 \cdot 2^h \][/tex]
First, divide both sides by 100:
[tex]\[ 64 = 2^h \][/tex]
Using properties of exponents, we know:
[tex]\[ 2^6 = 64 \][/tex]
Thus,
[tex]\[ h = 6 \][/tex]
6. Final Exponential Equation:
Based on the calculated values, the exponential equation for the growth of the bacterial cells, accounting for the given values, is:
[tex]\[ y = 100 \cdot 2^x \][/tex]
Specifically, for [tex]\( h = 6 \)[/tex]:
[tex]\[ 6400 = 100 \cdot 2^6 \][/tex]
Therefore, this detailed, step-by-step solution shows that the values for [tex]\( A, b, h \)[/tex], and [tex]\( y \)[/tex] are 100, 2, 6, and 6400, respectively, in the context of exponential growth of bacterial cells.
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