Question: Question 12
The spread of a disease can be modeled as $N = 200\sqrt{t}$, where $N$ is the number of infected people, and $t$ is time (in days).
How long will it take until the number of infected people reaches 1,400?
1. **State the problem:** We are given the model $$N = 200\sqrt{t}$$ where $N$ is the number of infected people and $t$ is time in days.
2. **Goal:** Find $t$ when $N = 1400$.
3. **Set up the equation:** Substitute $N = 1400$ into the formula:
$$1400 = 200\sqrt{t}$$
4. **Isolate $\sqrt{t}$:** Divide both sides by 200:
$$\frac{1400}{200} = \sqrt{t}$$
$$\cancel{\frac{1400}{\cancel{200}}} = \sqrt{t}$$
Simplify:
$$7 = \sqrt{t}$$
5. **Solve for $t$:** Square both sides to remove the square root:
$$7^2 = (\sqrt{t})^2$$
$$49 = t$$
6. **Answer:** It will take **49 days** for the number of infected people to reach 1,400.