Answer :
Let's solve the problem step by step to find the doubling time of the bacteria, rounding to the nearest minute.
### Step 1: Understand the Problem
We are given a bacterial culture that grows exponentially. The number of bacteria after a time [tex]\( t \)[/tex] can be expressed as:
[tex]\[ A = A_0 \times (C)^{\frac{1}{p}} \][/tex]
Where:
- [tex]\( A \)[/tex] is the number of bacteria at time [tex]\( t \)[/tex].
- [tex]\( A_0 \)[/tex] is the initial number of bacteria.
- [tex]\( C \)[/tex] is the growth constant.
- [tex]\( p \)[/tex] is the period impacting growth, often in terms of time units.
### Step 2: Identify Given Values
- Initial number of bacteria, [tex]\( A_0 = 3000 \)[/tex].
- Number of bacteria after 3 hours, [tex]\( A = 48000 \)[/tex].
### Step 3: Find the Growth Constant [tex]\( C \)[/tex]
Expressing the relationship at 3 hours:
[tex]\[ 48000 = 3000 \times (C)^{\frac{1}{3}} \][/tex]
To find [tex]\( C \)[/tex], rearrange the formula:
[tex]\[ C = \left(\frac{48000}{3000}\right)^3 \][/tex]
Calculate:
[tex]\[ \frac{48000}{3000} = 16 \][/tex]
[tex]\[ C = 16^{1/3} \approx 2.52 \][/tex]
### Step 4: Determine the Doubling Time
The doubling time [tex]\( T \)[/tex] satisfies the equation:
[tex]\[ 2 = (C)^{T/\text{hour}} \][/tex]
Convert the formula into a form we can solve:
[tex]\[ T = \text{hour} \cdot \frac{\log(2)}{\log(C)} \][/tex]
Calculate:
Using [tex]\( C \approx 2.52 \)[/tex]:
[tex]\[ T = 60 \cdot \frac{\log(2)}{\log(2.52)} \][/tex]
Perform the calculation:
[tex]\[ T \approx 45 \text{ minutes} \][/tex]
### Final Answer
The doubling time of the bacteria, rounded to the nearest minute, is 45 minutes.
### Step 1: Understand the Problem
We are given a bacterial culture that grows exponentially. The number of bacteria after a time [tex]\( t \)[/tex] can be expressed as:
[tex]\[ A = A_0 \times (C)^{\frac{1}{p}} \][/tex]
Where:
- [tex]\( A \)[/tex] is the number of bacteria at time [tex]\( t \)[/tex].
- [tex]\( A_0 \)[/tex] is the initial number of bacteria.
- [tex]\( C \)[/tex] is the growth constant.
- [tex]\( p \)[/tex] is the period impacting growth, often in terms of time units.
### Step 2: Identify Given Values
- Initial number of bacteria, [tex]\( A_0 = 3000 \)[/tex].
- Number of bacteria after 3 hours, [tex]\( A = 48000 \)[/tex].
### Step 3: Find the Growth Constant [tex]\( C \)[/tex]
Expressing the relationship at 3 hours:
[tex]\[ 48000 = 3000 \times (C)^{\frac{1}{3}} \][/tex]
To find [tex]\( C \)[/tex], rearrange the formula:
[tex]\[ C = \left(\frac{48000}{3000}\right)^3 \][/tex]
Calculate:
[tex]\[ \frac{48000}{3000} = 16 \][/tex]
[tex]\[ C = 16^{1/3} \approx 2.52 \][/tex]
### Step 4: Determine the Doubling Time
The doubling time [tex]\( T \)[/tex] satisfies the equation:
[tex]\[ 2 = (C)^{T/\text{hour}} \][/tex]
Convert the formula into a form we can solve:
[tex]\[ T = \text{hour} \cdot \frac{\log(2)}{\log(C)} \][/tex]
Calculate:
Using [tex]\( C \approx 2.52 \)[/tex]:
[tex]\[ T = 60 \cdot \frac{\log(2)}{\log(2.52)} \][/tex]
Perform the calculation:
[tex]\[ T \approx 45 \text{ minutes} \][/tex]
### Final Answer
The doubling time of the bacteria, rounded to the nearest minute, is 45 minutes.
