Problem 81 - College Algebra

Question

For the following exercises, express a rational function that describes the situation. A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after \(t\) minutes.

Step-by-Step Solution

Answer

The concentration function is \( C(t) = \frac{8 + 2t}{300 + 20t} \).

Step 1: Identify Initial Conditions

Initially, the tank contains 300 gallons of water and 8 pounds of sugar. We begin by recording these initial values for water and sugar in terms of gallons and pounds, respectively.

Step 2: Formulate the Water Function

The water is being poured into the tank at a rate of 20 gallons per minute. Thus, the amount of water in the tank after \( t \) minutes is given by the function \( W(t) = 300 + 20t \), where 300 represents the initial gallons, and 20t accounts for the increase over time.

Step 3: Formulate the Sugar Function

Similarly, sugar is poured into the tank at a rate of 2 pounds per minute. This gives the sugar function as \( S(t) = 8 + 2t \), where 8 is the initial pounds, and 2t represents the increase over time.

Step 4: Define the Concentration Function

Concentration is defined as the amount of sugar per the amount of water. Thus, the concentration function in terms of \( t \) is given by \( C(t) = \frac{S(t)}{W(t)} = \frac{8 + 2t}{300 + 20t} \).

Step 5: Simplify the Rational Function

The concentration function \( C(t) = \frac{8 + 2t}{300 + 20t} \) is already in its simplest form given the parameters of the problem.

Key Concepts

Concentration Calculation

Concentration is an important concept in problems involving mixtures. It tells us how much of a substance (in this case, sugar) is present in a mixture of another substance (here, water). Calculating concentration involves determining how many pounds of sugar are in a particular number of gallons of water. To find this, we use the formula for concentration:

• Concentration = (Amount of sugar) / (Amount of water)

In the given exercise, the amount of sugar and water can change over time as sugar and water are continuously being added. The rational function we use to determine concentration shows this relationship over time, denoted by:\[C(t) = \frac{8 + 2t}{300 + 20t}\]Here, \(8 + 2t\) represents the total sugar, and \(300 + 20t\) the total water. This setup allows us to understand how sugar concentration evolves as time goes on.

Rate of Change

The rate of change is a concept used to describe how quickly something is increasing or decreasing over time. In this exercise, it applies to both the water and the sugar being added continuously to the mixing tank. Recognizing the rate at which each element changes is crucial for forming accurate mathematical models.

• Water is being added at 20 gallons per minute.
• Sugar is being added at 2 pounds per minute.

These rates of change imply that every minute, 20 more gallons of water and 2 pounds of sugar are added. This constant addition influences the formulas for water and sugar content, \(W(t) = 300 + 20t\) and \(S(t) = 8 + 2t\), by representing continuous input over time. Understanding these rates helps visualize the dynamic nature of mixtures in real-time scenarios.

Initial Conditions

Initial conditions serve as the starting point for many mathematical models. They represent the state of the system at the beginning of observation. In our scenario, we begin with:

• 300 gallons of water
• 8 pounds of sugar

These values are crucial as they provide the baseline upon which all future calculations are based. The initial conditions are plugged into equations as constants, giving us a foundation to work from. Recognizing and clearly defining these conditions at the start ensures that the mathematical model accurately portrays the real-world situation from its inception.

Mathematical Modeling

Mathematical modeling involves creating equations or functions to represent real-world situations. This exercise presents a classic example of using mathematical tools to describe changes in a dynamic system, like the mixing tank. Models allow us to predict future behavior under various conditions, which is invaluable for planning and analysis.In this problem:

• We create separate functions for water (\(W(t) = 300 + 20t\)) and sugar (\(S(t) = 8 + 2t\)).
• These are combined into one function to reflect the concentration of sugar over time.\[C(t) = \frac{S(t)}{W(t)} = \frac{8 + 2t}{300 + 20t}\]

The concentration function helps us understand how adding substances at specific rates impacts the mixture. Mathematical models simplify complex real scenarios, providing clear insights into how elements interact and change over time. This way, they become powerful tools for solving practical problems.