Thank you for the clarification. Based on the correct equation:

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

I'll walk through the problem again to ensure that we correctly solve for $t$ when the concentration $c(t) = 6$ kg/m³, and then discuss the long-term behavior of the concentration.

Part (a):

Set $c(t) = 6$ kg/m³:

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

Isolate the exponential term:

$90,000 \left(\frac{1}{10,000}\right)^{\frac{t}{3}} = 3$

Simplify and solve for $t$:

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

Taking the natural logarithm of both sides:

$\frac{t}{3} \ln\left(\frac{1}{10,000}\right) = \ln\left(\frac{1}{30,000}\right)$

Now solve for $t$:

$t = 3 \times \frac{\ln\left(\frac{1}{30,000}\right)}{\ln\left(\frac{1}{10,000}\right)}$

Part (b):

As $t$ increases indefinitely, the term $\left(\frac{1}{10,000}\right)^{\frac{t}{3}}$ will tend toward zero, causing the concentration $c(t)$ to approach 9 kg/m³.

I'll now recalculate $t$ using this equation.### Part (a): The time $t$ when the concentration of the pollutant in the pond will reach 6 kg/m³ is approximately 3.36 minutes.

Part (b):

As time $t$ increases, the concentration $c(t)$ will approach 9 kg/m³ because the exponential term $\left(\frac{1}{10,000}\right)^{\frac{t}{3}}$ tends toward zero over time.

Would you like any further details or have any other questions?

Here are some related questions to consider:

1. How would the result change if the initial concentration was less than 9 kg/m³?
2. What is the impact on the time $t$ if the concentration required is 7 kg/m³ instead of 6 kg/m³?
3. How does the value $90,000$ in the equation affect the concentration behavior over time?
4. What are the practical implications of the concentration asymptotically approaching 9 kg/m³?
5. How would the result be different if the time factor $\frac{t}{3}$ was instead $\frac{t}{4}$?
6. Could this model apply to a scenario where the pollutant is being removed rather than added?
7. What happens to the concentration $c(t)$ if $t = 0$?
8. How can we graphically represent the concentration $c(t)$ over time?

Tip: Understanding the asymptotic behavior of functions is crucial in predicting the long-term effects of processes modeled by exponential functions.