Answer :
To model the number of bacteria in a colony that begins with a single cell and doubles in size every hour, we use an exponential function.
### Step-by-Step Solution
1. Identify the Initial Quantity (A):
- The initial number of bacteria is [tex]\( A = 1 \)[/tex]. This is because the colony starts with a single bacterial cell.
2. Determine the Base of the Exponential Function (b):
- The bacteria colony doubles in size every hour. This means the growth rate is exponential with a base [tex]\( b = 2 \)[/tex].
3. Write the Exponential Expression:
- The general form of an exponential function for growth is [tex]\( A \cdot (b^x) \)[/tex], where:
- [tex]\( A \)[/tex] is the initial amount,
- [tex]\( b \)[/tex] is the growth factor per time period (in this case, per hour),
- [tex]\( x \)[/tex] is the number of hours.
Given these values:
- [tex]\( A = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
We can write the expression to model the number of bacteria after [tex]\( x \)[/tex] hours as:
[tex]\[ 1 \cdot (2^x) \][/tex]
Therefore, the exponential expression is:
[tex]\[ A \cdot (b^x) = 1 \cdot (2^x) \][/tex]
### Result:
So, the exponential expression to model the bacterial count after [tex]\( x \)[/tex] hours is:
[tex]\[ 1 \cdot (2^x) \][/tex]
### Final Form:
To summarize, the values are:
- [tex]\( A = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
Thus, the complete exponential expression is:
[tex]\[ 1 \cdot (2^x) \][/tex]
Thank you for following along with this explanation. If you have further questions or need clarification on any step, feel free to ask!
### Step-by-Step Solution
1. Identify the Initial Quantity (A):
- The initial number of bacteria is [tex]\( A = 1 \)[/tex]. This is because the colony starts with a single bacterial cell.
2. Determine the Base of the Exponential Function (b):
- The bacteria colony doubles in size every hour. This means the growth rate is exponential with a base [tex]\( b = 2 \)[/tex].
3. Write the Exponential Expression:
- The general form of an exponential function for growth is [tex]\( A \cdot (b^x) \)[/tex], where:
- [tex]\( A \)[/tex] is the initial amount,
- [tex]\( b \)[/tex] is the growth factor per time period (in this case, per hour),
- [tex]\( x \)[/tex] is the number of hours.
Given these values:
- [tex]\( A = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
We can write the expression to model the number of bacteria after [tex]\( x \)[/tex] hours as:
[tex]\[ 1 \cdot (2^x) \][/tex]
Therefore, the exponential expression is:
[tex]\[ A \cdot (b^x) = 1 \cdot (2^x) \][/tex]
### Result:
So, the exponential expression to model the bacterial count after [tex]\( x \)[/tex] hours is:
[tex]\[ 1 \cdot (2^x) \][/tex]
### Final Form:
To summarize, the values are:
- [tex]\( A = 1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
Thus, the complete exponential expression is:
[tex]\[ 1 \cdot (2^x) \][/tex]
Thank you for following along with this explanation. If you have further questions or need clarification on any step, feel free to ask!
