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The equation is c(t) 𝑐(𝑡) = 9 − 90 000 ( 1 ), where c is

 10 000 􏰂3􏰁 a) When will the concentration of pollutant in the pond reach 6 𝑘𝑔/𝑚3 ?

measured in kilograms per cubic metre.

b) What will happen to the concentration of pollutant over time?

[2 marks]

Thank you for the clarification. The equation provided now is:

\[
c(t) = 9 - 90,000 \left(\frac{1}{10,000 + t}\right)^3
\]

### Part (a)
**Question**: When will the concentration of pollutant in the pond reach \( 6 \, \text{kg/m}^3 \)?

To solve for \( t \), set \( c(t) = 6 \) and solve the equation:

\[
6 = 9 - 90,000 \left(\frac{1}{10,000 + t}\right)^3
\]

Subtract 9 from both sides:

\[
-3 = -90,000 \left(\frac{1}{10,000 + t}\right)^3
\]

Divide both sides by \(-90,000\):

\[
\frac{1}{30,000} = \left(\frac{1}{10,000 + t}\right)^3
\]

Take the cube root of both sides:

\[
\frac{1}{\sqrt[3]{30,000}} = \frac{1}{10,000 + t}
\]

Multiply both sides by \( 10,000 + t \):

\[
10,000 + t = \sqrt[3]{30,000}
\]

Subtract 10,000 from both sides:

\[
t = \sqrt[3]{30,000} - 10,000
\]

Finally, compute \( t \):

\[
t \approx 31.9 - 10,000
\]

\[
t \approx -9,968.1 \, \text{minutes}
\]

Since the calculated \( t \) is negative, this suggests that \( c(t) = 6 \, \text{kg/m}^3 \) never actually happens in the positive time domain, meaning the concentration does not reach 6 kg/m³ after the polluted water starts flowing into the pond.

### Part (b)
**Question**: What will happen to the concentration of pollutant over time?

As \( t \) increases to a very large value (approaches infinity), the term \( \frac{1}{10,000 + t} \) becomes very small because the denominator dominates, making the concentration \( c(t) \) approach:

\[
c(t) \to 9 \, \text{kg/m}^3
\]

Thus, the concentration will eventually stabilize at 9 kg/m³.

### Would you like more details or have any questions? 

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### Related Questions:
1. How does the initial condition of the pond (e.g., its size or initial concentration) affect this model?
2. What assumptions are made in this model, and how might they affect the results?
3. How would the model change if the rate of pollution was not constant?
4. Could there be a scenario where the concentration would decrease instead of increasing?
5. How does the parameter 90,000 affect the overall concentration behavior?
6. What would happen if the pond had an outflow that reduced the concentration over time?
7. How can this problem be solved using a numerical method?
8. Is it possible to generalize this model to other pollutants or substances?

**Tip**: When solving equations with possible physical implications, always check if the solutions are feasible within the context. For example, negative time might indicate the event cannot occur.

Explore how to solve the concentration of pollutant over time with the equation c(t) = 9 - 90,000 (1 / (10,000 + t)^3). Find out when the concentration reaches 6 kg/m³ and understand the long-term behavior of the concentration as time increases. Suitable for advanced high school students and beyond.

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