Exponential Growth 1B5E30

1. **State the problem:** We are given the number of infections for weeks 0 to 3 and told the infections grow at a rate of 50% per week. We want to find the number of infections at week 6 assuming exponential growth. 2. **Formula for exponential growth:** The general formula is $$N(t) = N_0 \times (1 + r)^t$$ where: - $N(t)$ is the number of infections at week $t$, - $N_0$ is the initial number of infections, - $r$ is the growth rate per time period (as a decimal), - $t$ is the number of weeks after the initial time. 3. **Identify given values:** - $N_0 = 40$ (infections at week 0), - $r = 0.50$ (50% growth rate), - $t = 6$ (week 6). 4. **Calculate $N(6)$:** $$N(6) = 40 \times (1 + 0.50)^6 = 40 \times (1.5)^6$$ 5. **Calculate $(1.5)^6$ step-by-step:** - $(1.5)^2 = 2.25$ - $(1.5)^3 = 1.5 \times 2.25 = 3.375$ - $(1.5)^4 = 1.5 \times 3.375 = 5.0625$ - $(1.5)^5 = 1.5 \times 5.0625 = 7.59375$ - $(1.5)^6 = 1.5 \times 7.59375 = 11.390625$ 6. **Multiply by initial infections:** $$N(6) = 40 \times 11.390625 = 455.625$$ 7. **Interpretation:** Since the number of infections must be a whole number, we round to the nearest whole number: $$\boxed{456}$$ infections at week 6. Thus, if the infections continue to grow exponentially at 50% per week, there will be approximately 456 infections in week 6.