Problem 35 - Precalculus Mathematics for Calculus

Question

(a) A tank contains \(5000 \mathrm{L}\) of pure water. Brine that contains \(30 \mathrm{g}\) of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after \(t\) minutes (in grams per liter) is $$C(t)=\frac{30 t}{200+t}$$ (b) What happens to the concentration as \(t \rightarrow \infty ?\)

Step-by-Step Solution

Answer

As \( t \rightarrow \infty \), \( C(t) \) approaches 30 grams per liter.

Step 1: Understanding the problem

The tank initially contains 5000 liters of pure water and brine with 30 grams of salt per liter enters the tank at 25 liters per minute. We need to find the concentration of salt at any time \( t \), represented by \( C(t) = \frac{30t}{200+t} \). We also need to determine the behavior of this concentration as \( t \) approaches infinity.

Step 2: Determine Salt Added Over Time

The brine solution entering the tank contains 30 grams of salt per liter. Therefore, in one minute, \(25\) L/min enters the tank, adding \(25 \times 30 = 750\) grams of salt per minute. After \(t\) minutes, the total amount of salt added to the tank is \(750t\) grams.

Step 3: Determine Total Volume Over Time

The brine solution enters the tank at a rate of 25 L/min. Therefore, after \(t\) minutes, the amount of brine added is \(25t\) liters. Since the tank initially contained 5000 liters of water, the total volume of liquid in the tank after \(t\) minutes is \(5000 + 25t\) liters.

Step 4: Calculate Salt Concentration at Time t

The concentration of salt at time \(t\) is found by dividing the total salt mass \(750t\) by the total volume \(5000 + 25t\). Thus, \[ C(t) = \frac{750t}{5000 + 25t} = \frac{30t}{200+t} \] simplifying by factoring common terms.

Step 5: Analyze Behavior as t Approaches Infinity

To understand what happens as \( t \rightarrow \infty \), analyze the equation \( C(t) = \frac{30t}{200 + t} \). As \( t \) becomes very large, the \( 200 \) in the denominator becomes negligible, making \( C(t) \approx \frac{30t}{t} = 30 \). Thus, the concentration approaches 30 grams per liter.

Key Concepts

Brine Solution

Imagine you have a tank filled with pure water, and you start adding a special mix of water and salt called a "brine solution." This brine solution is not just any solution; it has a specific amount of salt. Here, the brine contains 30 grams of salt for every liter of water.

When this brine flows into the tank, it comes at a regular pace, like a steady stream from a faucet, at 25 liters every minute.

As the brine enters the tank, more and more salt makes its way into the water, changing the water from pure to salty gradually. Over time, the concentration or amount of salt in the tank starts to build up.

The challenge is to figure out exactly how salty the water becomes as time goes on. This concentration is what we calculate as a function of time, denoted as \(C(t)\).

Rate of Change

The rate of change in this context tells us how quickly the salt concentration in the tank is increasing. Every minute, 25 liters of brine enter the tank, bringing with it a total of 750 grams of salt per minute. This is because each liter carries 30 grams of salt.

• Total salt added in one minute = 25 liters × 30 grams/liter = 750 grams.
• After \(t\) minutes, total salt = 750 \(t\) grams.

For concentration, not only the amount of salt matters but also the size of the tank --- or more precisely, the total volume of water. With each passing minute, 25 more liters of liquid add to the initial 5000 liters of water. Thus, the total volume after \(t\) minutes becomes \(5000 + 25t\) liters.

To find how concentrated the salt is at any time \(t\), we use: \[ C(t) = \frac{750t}{5000 + 25t} = \frac{30t}{200 + t} \]

This equation gives us a clear picture of how the concentration of salt changes over time.

Long-term Behavior

So, what happens if we keep pouring this brine solution into the tank indefinitely? Let's dive into the long-term behavior of the salt concentration.

As \(t\) continues to grow and becomes a very large number, the smaller number 200 in our equation \(C(t) = \frac{30t}{200 + t}\) starts to become insignificant. The math behind this is simple.

Instead of the equation having a substantial addition in the denominator, it now looks more like \(\frac{30t}{t}\), because 200 is far smaller compared to \(t\).

Thus, this simplifies even further to simply 30:

• Long-term salt concentration = 30 grams per liter.

Therefore, as time goes on and on, the concentration stabilizes and approaches 30 grams per liter, which is the same concentration of salt in the incoming brine solution. This tells us that eventually, the tank achieves a balance where the incoming and existing concentrations of salt are matched.