Problem 535 - Precalculus

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 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 after \(t\) minutes is \(\frac{8 + 2t}{300 + 20t}\) pounds per gallon.

Step 1: Determine the initial amounts

First, identify the initial quantities: there are 300 gallons of water and 8 pounds of sugar in the tank.

Step 2: Determine the change over time

Each minute, 20 gallons of water and 2 pounds of sugar are added to the tank. Therefore, in \(t\) minutes, \(20t\) gallons of water and \(2t\) pounds of sugar are added.

Step 3: Formulate expressions for total amounts

The total volume of water in the tank after \(t\) minutes is given by \(300 + 20t\) gallons. The total amount of sugar is \(8 + 2t\) pounds.

Step 4: Define the concentration function

The concentration (in pounds per gallon) after \(t\) minutes can be expressed as the ratio of sugar to water: \(\frac{8 + 2t}{300 + 20t}\).

Key Concepts

Concentration in a Solution

When we talk about concentration in the context of chemistry or mixing problems, we are essentially discussing the amount of a certain substance (in this case, sugar) that is present in a mixture (here, water). Concentration is typically expressed as a ratio, such as pounds per gallon, which gives us a clear measure of how "strong" or "dilute" a solution is.

In the tank problem described, concentration plays a central role. We are tasked with finding the sugar concentration after a specified time while more water and sugar are being added. As the amount of both components changes over time, the concentration is not static and must be calculated as a function of time. This leads us to express the concentration as a rational function—which is simply the quotient of two polynomials. Here, concentration \( C(t) \) is defined as: \[ C(t) = \frac{8 + 2t}{300 + 20t} \] where the numerator represents the total amount of sugar and the denominator represents the total volume of the solution.

By understanding this function, we can evaluate how the concentration changes with each passing minute.

Pounds per Gallon - A Unit of Measurement

The unit "pounds per gallon" is used to measure concentration in this scenario, denoting how many pounds of sugar are present in each gallon of water. This unit makes it easier to grasp the concept of concentration as it directly relates the amount of solute (sugar) to the volume of the solvent (water).

In the given problem, initially, the tank contains 8 pounds of sugar mixed into 300 gallons of water. As more sugar and water are added, this ratio changes, and we must adjust our understanding of pounds per gallon over time.

• The numerator \(8 + 2t\) reflects the growing amount of sugar.
• The denominator \(300 + 20t\) represents the increasing water volume.

Calculating in pounds per gallon is helpful because it normalizes the concentration relative to the amount of solution, allowing us to see concentration changes directly related to the amount of liquid.

Understanding the Tank Problem

Tank problems, like the one provided, are common scenarios found in mathematics and engineering, where the behavior of mixtures is examined under dynamic conditions. These problems typically involve tracking changes in content or concentration over time, often requiring a methodical approach to find an exact solution.

In our scenario, the tank problem is a visual way to understand the continuous inflow of materials (water and sugar) and forces us to think about how these flows affect the concentration of sugar. As each minute passes, the tank incorporates more sugar and water at consistent rates, prompting us to use algebraic expressions to describe these changes. Consequently, rational functions serve as a tool to express the evolving concentrations, making these problems surprisingly applicable to real-life situations where rates and quantities shift seamlessly.

Sugar Solution Dynamics

The problem of the sugar solution involves observing how the sugar concentration changes as additional sugar and water mix continuously into the tank. This idea can easily be extended to real-world applications, such as in food processing or chemical solution management, where precise control of concentration is crucial.

Here, we are observing a simplified model where the input rates for both sugar and water are constant. It is essential to understand the dynamics:

• More sugar per minute increases the solution's sweetness and hence its concentration.
• Additional water dilutes the solution, reducing its concentration if not balanced by more sugar.

As you solve these types of problems, recognizing the balance between these two competing processes—adding sugar to sweeten, versus adding water to dilute—is key. It highlights the importance of the initial mixing and the constant feed rates, which allow for precise calculation of concentration changes over time. This knowledge is invaluable in any context where solutions or mixtures must be carefully controlled.