1. **State the problem:** We want to find the number of weeks $w$ after the beehive's establishment when the population $B$ reaches 720. 2. **Given equation:** The population is modeled by $$B = 360e^{0.03w}$$ 3. **Set up the equation to solve for $w$: $$720 = 360e^{0.03w}$$** 4. **Divide both sides by 360:** $$\frac{720}{360} = \frac{360e^{0.03w}}{360}$$ $$2 = e^{0.03w}$$ 5. **Take the natural logarithm of both sides:** $$\ln(2) = \ln\left(e^{0.03w}\right)$$ $$\ln(2) = 0.03w$$ 6. **Solve for $w$:** $$w = \frac{\ln(2)}{0.03}$$ 7. **Calculate the value:** $$w \approx \frac{0.6931}{0.03} \approx 23.1$$ 8. **Interpretation:** The population will reach 720 approximately 23 weeks after the beehive's establishment. **Final answer:** 23 weeks