1. **State the problem:** We are given the function $$f(t) = 40000 \cdot 2^{\frac{t}{790}}$$ which models the number of bacteria at time $$t$$ minutes. We need to find the time $$t$$ it takes for the population to double. 2. **Understand what doubling means:** Doubling means the population becomes twice the initial amount. The initial population at $$t=0$$ is $$f(0) = 40000$$. We want to find $$t$$ such that $$f(t) = 2 \times 40000 = 80000$$. 3. **Set up the equation:** $$ 40000 \cdot 2^{\frac{t}{790}} = 80000 $$ 4. **Divide both sides by 40000:** $$ \cancel{40000} \cdot 2^{\frac{t}{790}} = 2 \times \cancel{40000} $$ which simplifies to $$ 2^{\frac{t}{790}} = 2 $$ 5. **Rewrite the right side as a power of 2:** $$ 2^{\frac{t}{790}} = 2^1 $$ 6. **Since the bases are equal, set the exponents equal:** $$ \frac{t}{790} = 1 $$ 7. **Solve for $$t$$:** $$ t = 790 $$ **Answer:** It takes $$790$$ minutes for the bacteria population to double.