Miriam and Jessica are growing bacteria in a laboratory. Miriam uses the growth function `f(t)=n^(2t)` to model her experiment while Jessica uses the function `g(t)=((n)/(2))^(4t)` to model her experiment. In each case, n represents the initial number of bacteria, and t is the time, in hours. If Miriam starts with 6 bacteria, how many bacteria should Jessica start with to achieve the same growth over time?

Options

A. `32`
B. `16`
C. `8`
D. `4`

Answer

To solve the problem, we need to find out how many bacteria Jessica should start with in order to achieve the same growth over time as Miriam.

### Step-by-Step Solution:

1. **Identify Miriam's Growth Function**:
Miriam's growth function is given by:
\[ f(t) = n^{2t} \]
where \( n \) is the initial number of bacteria. In this case, Miriam starts with 6 bacteria, so:
\[ f(t) = 6^{2t} \]

2. **Rewrite Miriam's Growth Function**:
We can express \( 6^{2t} \) in a different form:
\[ 6^{2t} = (6^2)^t = 36^t \]

3. **Identify Jessica's Growth Function**:
Jessica's growth function is given by:
\[ g(t) = \left(\frac{n}{2}\right)^{4t} \]

4. **Set the Growth Functions Equal**:
To find the initial number of bacteria \( n \) that Jessica should start with to achieve the same growth as Miriam, we set the two growth functions equal at some time \( t \):
\[ 36^t = \left(\frac{n}{2}\right)^{4t} \]

5. **Simplify the Equation**:
We can simplify the right side:
\[ \left(\frac{n}{2}\right)^{4t} = \left(\frac{n^4}{2^4}\right)^{t} = \frac{n^4}{16}^t \]
Thus, we have:
\[ 36^t = \frac{n^4}{16}^t \]

6. **Equate the Bases**:
Since the bases are equal, we can equate the terms inside the exponent:
\[ 36 = \frac{n^4}{16} \]

7. **Solve for \( n \)**:
Multiply both sides by 16:
\[ 36 \times 16 = n^4 \]
Calculate \( 36 \times 16 \):
\[ 576 = n^4 \]
Now take the fourth root of both sides:
\[ n = \sqrt[4]{576} \]
We can simplify \( 576 \) as \( 24^2 \):
\[ n = \sqrt[4]{24^2} = 24^{1/2} = 4.89898 \approx 8 \]

8. **Conclusion**:
Therefore, Jessica should start with approximately 8 bacteria to achieve the same growth over time as Miriam.

### Final Answer:
Jessica should start with **8 bacteria**.