Problem 51 - Precalculus: Functions and Graphs

Question

Salt water of concentration 0.1 pound of salt per gallon flows into a large tank that initially contains 50 gallons of pure water. (a) If the flow rate of salt water into the tank is 5 gal/min, find the volume \(V(t)\) of water and the amount \(A(t)\) of salt in the tank after \(t\) minutes. (b) Find a formula for the salt concentration \(c(t)\) (in Ib/gal) after \(t\) minutes. (c) Discuss the variation of \(c(t)\) as \(t \rightarrow \infty\)

Step-by-Step Solution

Answer

(a) \( V(t) = 50 + 5t \), \( A(t) = 0.5t \). (b) \( c(t) = \frac{t}{10+t} \). (c) \( c(t) \rightarrow 1 \) as \( t \rightarrow \infty \).

Step 1: Determine the Volume Function

The volume of water in the tank initially is 50 gallons. With the entry of salt water at a rate of 5 gallons per minute, the volume of water in the tank at time \( t \) minutes is\[V(t) = 50 + 5t.\]

Step 2: Determine the Amount of Salt Function

Since the inflow of water is 0.1 pounds per gallon and the flow rate is 5 gallons per minute, the rate at which the salt is added is \( 0.1 \times 5 = 0.5 \) pounds per minute. Thus, the amount of salt in the tank after \( t \) minutes is\[A(t) = 0.5t.\]

Step 3: Calculate Salt Concentration Function

The concentration \( c(t) \) is given by the amount of salt per unit volume. Therefore,\[c(t) = \frac{A(t)}{V(t)} = \frac{0.5t}{50+5t} = \frac{0.5t}{5(10+t)} = \frac{t}{10+t}.\]

Step 4: Analyze Salt Concentration as Time Approaches Infinity

As \( t \rightarrow \infty \), the term \( 10 + t \) becomes dominated by \( t \), so:\[\lim_{{t \to \infty}} \frac{t}{10 + t} = 1.\]This means the concentration approaches 1 pound per gallon as time goes to infinity.

Key Concepts

Salt concentration

Salt concentration describes the amount of salt per unit volume in a solution. In this problem, the concentration is initially determined by the amount of salt entering the tank compared to the total volume of water. As water continuously flows into the tank, this concentration is constantly changing. The formula for salt concentration, after time \( t \) minutes, is derived from dividing the amount of salt \( A(t) \) by the volume of water \( V(t) \). This gives us:

• \( c(t) = \frac{t}{10+t} \)
• Here, \( t \) represents time, making it explicitly time-dependent.
• Although the inflow rate affects how quickly salt concentration changes, the concentration should stabilize over time.

Volume of water

The volume of water in the tank increases over time as salt water continuously flows in. Initially, the tank contains 50 gallons of pure water. Salt water is added at a rate of 5 gallons per minute. This forms a linear relationship for the volume with respect to time:\[V(t) = 50 + 5t.\]

• The initial volume is simply the starting condition before any salt water addition.
• The linear term \( 5t \) reflects the constant inflow rate of salt water.
• Each minute, an additional 5 gallons increases the total volume, modifying the tank’s salt concentration.

Understanding this growth in volume helps in figuring out how diluted or concentrated the salt solution will become as time progresses.

Rate of change

The rate of change is crucial in understanding how various quantities in a problem evolve over time. Here we talk about two main rates: the salt inflow rate and the water inflow rate.

• The **inflow rate of salt** is 0.1 pounds for each of the 5 gallons flowing in, resulting in 0.5 pounds of salt per minute.
• Simultaneously, the **water inflow rate** is 5 gallons per minute, indicating a constant increase in volume.

These rates govern both the quantity of salt and the changing volume within the tank. Monitoring how they interact allows prediction of salt concentration dynamics and emphasizes how different components of the system rely on each other's changes.

Limit of a function

When examining long-term behavior in mathematics, the concept of the limit becomes invaluable. In this scenario, it looks at what happens to the salt concentration \( c(t) \) as time \( t \) approaches infinity.

• Given \( c(t) = \frac{t}{10+t} \), as \( t \) grows, \(10+t\) simplifies towards just \( t \), leading to \( c(t) \) approaching 1.
• The limit \( \lim_{t \to \infty} \frac{t}{10+t} = 1 \) captures the idea that the concentration stabilizes, with 1 pound per gallon being the reaching equilibrium.

This aspect of limits helps predict behavior in complex systems over extended periods.

Inflow and mixing problems

Inflow and mixing problems are central to applied differential equations and involve substances entering and blending within a system.

• These systems typically consider the **balance and behavior of incoming substances** against existing ones, factoring in flow rates, initial amounts, and volume changes.
• In such problems, differential equations play a crucial role to express how various quantities evolve over time.
• The salt-water tank exercise serves as a classic example where it's essential to **track how a solution's composition changes** over time, especially with continuous inputs.

These problems teach the practical application of calculus, particularly in real-world engineering, environmental science, and chemistry contexts.