Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Salt concentration Salt water of concentration 0.1 pound of salt per gallon flows into a large tank that initially conains 50 gallons of pure water. (a) If the flow rate of salt water into the tank is \(5 \mathrm{gal} / \mathrm{min}\), find the volume \(V(t)\) of water and the amount \(A(t)\) of salt in the tank after \(t\) minutes.

Step-by-Step Solution

Answer

The volume of water, \(V(t)\), is \(50 + 5t\) gallons, and the amount of salt, \(A(t)\), is \(0.5t\) pounds after \(t\) minutes.

Step 1: Determine the Volume of Water Over Time

The water flows into the tank at a rate of 5 gallons per minute. Initially, the tank has 50 gallons of pure water. Hence, the volume of water in the tank after \(t\) minutes is described by:\[ V(t) = 50 + 5t \]This accounts for the initial 50 gallons and the additional 5 gallons of water per minute.

Step 2: Calculate the Rate of Salt Flow into the Tank

Salt water with a concentration of 0.1 pound of salt per gallon flows into the tank. Hence, the rate at which salt flows into the tank is given by:\[ \text{Salt flow rate} = 0.1 \times 5 = 0.5 \text{ pounds per minute} \]This means 0.5 pounds of salt enter the tank every minute.

Step 3: Determine the Amount of Salt Over Time

Since the initial amount of salt in the tank is zero and salt is flowing in at a constant rate, the amount of salt after \(t\) minutes is:\[ A(t) = 0.5t \]This equation reflects the accumulation of salt over time with no removal process involved.

Step 4: Express the Final Answers

Using the expressions from Step 1 and Step 3, the volume of water and amount of salt in the tank after \(t\) minutes are:- Volume of water, \( V(t) = 50 + 5t \)- Amount of salt, \( A(t) = 0.5t \)

Key Concepts

Understanding Salt Concentration

In problems involving tanks and mixtures, understanding how concentration works is crucial. Salt concentration tells us how much salt is present in a specific amount of liquid, usually expressed in pounds per gallon in these exercises. For instance, a salt water solution with a concentration of 0.1 pounds per gallon means that every single gallon of this solution contains 0.1 pounds of salt.
This concept is foundational, as it allows you to calculate how much salt enters a tank over time, given the flow rate of the solution. Once you grasp the idea of concentration, you can easily find the total amount of salt by multiplying the concentration by the volume entering the system.

Flow Rate and its Impact

Flow rate refers to the volume of fluid entering or exiting a system per unit of time, usually measured in gallons per minute in these contexts. In the exercise, the flow rate is 5 gallons per minute, which indicates how fast the tank is filling.
This rate directly affects the solution's properties over time. For example, in our problem, this consistent rate helps determine the increase in water volume and, given a constant salt concentration, the rate of salt addition.
Understanding flow rate is essential, as it helos solve how quickly a mixture changes in a tank-driven process. It acts as a bridge between the concentration information and the resulting changes in the tank's contents.

Calculating Water Volume in Tanks

The volume of water in a tank over time is impacted by the initial amount and the continuous addition of liquid. Initially, our tank holds 50 gallons. Over time, additional water is added at 5 gallons per minute. Therefore, after any minute, you can calculate the total water volume as the sum of the starting volume and the added volume: \[ V(t) = 50 + 5t \]
This equation clearly shows how volume changes uniformly with time. It's a linear relationship, meaning it increases steadily, which simplifies calculations for future states of the system.

• Initial Water Volume: 50 gallons
• Flow Rate: 5 gallons/minute
• Volume After t Minutes: 50 gallons + 5 gallons/minute × t minutes

Exploring Tanks and Mixtures

Tanks and mixtures are common contexts in fluid dynamics and chemistry problems, where disparate substances mix over time. These exercises often involve understanding how components like salt, water, or other substances interact dynamically.
In our scenario, a salt solution flows into a tank containing pure water. As time passes, the salt mixes uniformly, changing the solution's composition.
Key concepts include:

• Initial Conditions: Knowing what the tank starts with, such as any existing liquid or solute amount.
• Constant Rates: Whether flow rates or removal rates affect the system uniformly.
• Accumulative Effects: Recognizing that some compounds or volumes will increase over time.

This section helps you connect real-world processes to mathematical models, enabling you to predict behaviors inside tanks accurately.