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The problem you're dealing with concerns the control of algae proliferation in a lake. Scientists are using a treatment to manage the spread of algae, and they are modeling the area \( A \) (in square yards) of the lake that is covered by algae using the equation:

\[
A = 240 \cdot \left( \frac{1}{3} \right)^t
\]

where \( t \) is the time measured in weeks.

Let's break down the equation and answer the questions provided:

### 1. ¿Qué nos dice el 240 acerca de las algas?
The number 240 represents the initial area covered by algae in the lake before the treatment begins. This means that when \( t = 0 \), the algae cover an area of 240 square yards. This is the starting point of the algae-covered area, assuming no treatment has yet taken effect.

### 2. ¿Qué nos dice el \( \frac{1}{3} \)?
The fraction \( \frac{1}{3} \) represents the rate at which the algae-covered area decreases each week. Specifically, for each week that passes (each unit of \( t \)), the area is reduced to one-third of the area it covered the previous week. This shows that the treatment is effective, reducing the algae by a factor of \( \frac{1}{3} \) every week.

Would you like a more detailed explanation of how the area decreases over time, or any further clarification on these concepts?

---

Here are five related questions to deepen your understanding:

1. How would the equation change if the initial area covered by algae was 300 square yards?
2. What would happen to the algae-covered area if the treatment was less effective, say reducing the area by \( \frac{1}{2} \) each week instead of \( \frac{1}{3} \)?
3. After how many weeks would the area of algae be reduced to less than 10 square yards?
4. What is the area covered by algae after 2 weeks of treatment?
5. How does the area covered by algae behave over a long period (as \( t \to \infty \))?

**Tip**: To model exponential decay like this, always remember that the base (here \( \frac{1}{3} \)) represents how much remains after each time unit (in this case, weeks).

En la ecuación, ¿qué nos dice el 240 acerca de las algas? ¿Qué nos dice el 1/3?

Explore how scientists use exponential decay to control algae proliferation in a lake. Learn how the equation A = 240 * (1/3)^t explains the initial algae coverage and weekly decay rate. Suitable for students in Grades 8-10.

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