1. **Problem statement:** We have a colony of 100 bacteria that triples every hour. We want to find the population after a certain time. 2. **Formula:** The population after time $t$ hours is given by $$P(t) = P_0 \times 3^t$$ where $P_0 = 100$ is the initial population. 3. **Important rule:** Since the population triples every hour, the growth is exponential with base 3. 4. **Calculate for 4 hours:** $$P(4) = 100 \times 3^4 = 100 \times 81 = 8100$$ 5. **Calculate for 90 minutes (1.5 hours):** $$P(1.5) = 100 \times 3^{1.5} = 100 \times 3^{\frac{3}{2}} = 100 \times 3^{1} \times 3^{0.5} = 100 \times 3 \times \sqrt{3} = 300 \times 1.732 = 519.6$$ 6. **Calculate for 1/2 hour (0.5 hours):** $$P(0.5) = 100 \times 3^{0.5} = 100 \times \sqrt{3} = 100 \times 1.732 = 173.2$$ **Final answers:** - After 4 hours: 8100 bacteria - After 90 minutes: approximately 520 bacteria - After 1/2 hour: approximately 173 bacteria