Advanced functions (precalculus)

[MartynaJ]

Homework Statement

After a train derailment in Northern Ontario, the concentration, C, in grams per litre, of a pollutant after t minutes in a 5 000 000 L pond can be modelled by the function C(t)=30t/(200 000+t) where a pollutant concentration of 30 g/L flows into the pond at a rate of 25 L/min.
1) When a concentration level of 0.05 g/L in the pond is reached, the fish stock will be irreversibly damaged. When will this occur?
2) What happens to the concentration over time?

Relevant Equations

See above please

This is my attempt so far:

1. $0.05=\frac{30t}{200000+t}$ then I solved for t. And I got 333.88 min. I feel like this is way too simple of a solution and I didn't use all of what's given in the problem.
2. For part 2 of the problem it asks, what happens to the concentration over time. I tried to graph the equation and my graphs appears to have a slope of zero. But if I divide the numerator and denominator by time, I get: $C(t)=\frac{30}{\frac{200000}{t}+1}$, then making t--> infinity, the 200000/t will approach zero and the concentration will be 30.

Please let me know what I am doing wrong.
Thanks

 

[Office_Shredder]

You probably feel like you haven't used everything in the problem because you're wondering where e.g. the 5000000 L went, or the 25 L/min. They're actually secretly in the C(t) function, if you try to derive it yourself you'll see they basically canceled a 25 from the numerator and denominator.

I think both your answers look correct. It might be worth trying to understand why the answer to part b is 30 g/L beyond just the math. Are you able to describe it in words?

 

[MartynaJ]

You probably feel like you haven't used everything in the problem because you're wondering where e.g. the 5000000 L went, or the 25 L/min. They're actually secretly in the C(t) function, if you try to derive it yourself you'll see they basically canceled a 25 from the numerator and denominator.

I think both your answers look correct. It might be worth trying to understand why the answer to part b is 30 g/L beyond just the math. Are you able to describe it in words?

Yes exactly my point... for the first part, I didn't really use much of what was given in the problem. And for the second part I am not sure id I am able to explain in words why it's 30. I am also not too sure if $\frac{200000}{t}$ approaches zero as $t$ approaches infinity just because 200000 is a large number in itself...

 

[Office_Shredder]

200,000 is small compared to, say 200000000000000000. Which as t goes to infinity it will eventually get larger than. For any choice of constant c, c/t goes to 0 as t goes to infinity.

For why the last part is 30. You start with some finite amount of pure water. You pour in water that's polluted at a concentration of 30g/L. The pollution level of the whole thing is going to be some average of the 0g / L clean water and the 30g/L polluted water, depending on how much of the water is clean vs dirty. As time goes to infinity, we keep adding more and more polluted water, so the amount of clean water is negligible in size compared to the polluted water.