A tank contains 2000 L of pure water. Brine that contains 15 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. The concentration of salt after t minutes (in grams per liter) is C(t) = 15t/80 + t. As t → ∞, what does the concentration approach?
clearly you must have meant
C(t)=15t/(80+t) which --> 15
I am troubled by the fact that the equation for C(t) does not reflect the data given in the words. Also, at some point the tank will overflow, so having t unbounded is not realistic.
Well, when it comes to concentration, it's all about staying salty. Just kidding! Let's break it down.
As t approaches infinity, we're looking at what happens in the long run. The concentration function is C(t) = 15t/80 + t.
Now, if we divide both terms by t, we get C(t) = 15/80 + 1. As t gets larger and larger, the t term becomes less significant, almost like a clown's importance at a formal dinner party. So we are left with C(t) = 15/80 + 1.
Simplifying further, we find that C(t) = 15/80 + 1 = 95/80.
So, as t approaches infinity, the concentration approaches a clown-approved value of 95/80 grams per liter.
To find the concentration of salt as t approaches infinity, we need to evaluate the limit of C(t) as t approaches infinity.
The expression for C(t) is:
C(t) = 15t/80 + t
As t approaches infinity, the term 15t/80 becomes negligible compared to t. This is because as t gets larger and larger, the term 15t/80 becomes relatively smaller compared to t.
Therefore, the expression simplifies to:
C(t) ≈ t
As t approaches infinity, the concentration of salt approaches infinity as well.
So, the concentration of salt approaches infinity as t → ∞.
To find the concentration of salt as t approaches infinity, we can take the limit of the given concentration function C(t) as t goes to infinity.
The concentration function is given by:
C(t) = 15t/80 + t
As t approaches infinity, we can ignore the term 15t/80, because compared to the term t, it becomes negligible.
So, the concentration as t approaches infinity is simply the value of t:
C(t) = t
Therefore, the concentration of salt as t approaches infinity is t, which means it will keep increasing without bound as time goes on.