Question: User: User: A bacteria culture starts with $250$ bacteria and doubles every $3$ hours. After $15$ hours, how many bacteria will there be in the culture?
1. **State the problem.**
We start with $250$ bacteria, and the culture doubles every $3$ hours.
We want to know how many bacteria there are after $15$ hours.
2. **Choose the exponential growth formula.**
For doubling growth, the formula is
$$N=N_0\cdot 2^{t/d}$$
where $N_0$ is the starting amount, $t$ is the time passed, and $d$ is the doubling time.
3. **Substitute the values.**
Here, $N_0=250$, $t=15$, and $d=3$.
So,
$$N=250\cdot 2^{15/3}$$
4. **Simplify the exponent.**
$$15/3=5$$
So the expression becomes
$$N=250\cdot 2^5$$
5. **Evaluate the power.**
$$2^5=32$$
So now we have
$$N=250\cdot 32$$
6. **Multiply.**
$$250\cdot 32=8000$$
7. **Final answer.**
After $15$ hours, there will be $8000$ bacteria in the culture.