Problem 80 - College Algebra

Question

For the following exercises, express a rational function that describes the situation. A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after \(t\) minutes.

Step-by-Step Solution

Answer

Concentration is \( C(t) = \frac{10 + 3t}{200 + 10t} \).

Step 1: Define the Function Variables

Let's define the necessary variables for the problem. Let \( S(t) \) represent the amount of sugar in pounds in the tank at minute \( t \), and \( V(t) \) represent the volume of water in gallons at minute \( t \).

Step 2: Initial Values Setup

Initially, at \( t = 0 \), the tank has 10 pounds of sugar and 200 gallons of water. Therefore, \( S(0) = 10 \) pounds and \( V(0) = 200 \) gallons.

Step 3: Model the Changes Over Time

Every minute, 3 pounds of sugar are added, so the sugar function is \( S(t) = 10 + 3t \). Simultaneously, 10 gallons of water are added per minute, so the volume function is \( V(t) = 200 + 10t \).

Step 4: Express the Concentration Function

The concentration of sugar at time \( t \) is the amount of sugar divided by the volume of water in the tank. This is expressed as:\[C(t) = \frac{S(t)}{V(t)} = \frac{10 + 3t}{200 + 10t}\]

Step 5: Conclusion

The function \( C(t) = \frac{10 + 3t}{200 + 10t} \) describes the concentration of sugar in pounds per gallon in the tank at any time \( t \).

Key Concepts

Concentration

In the context of the exercise, concentration refers to the amount of sugar compared to the volume of water in the tank. It’s often expressed as pounds of sugar per gallon of water. Understanding concentration is crucial because it tells us how much sugar is in each gallon of the solution. This information is essential for creating consistent mixtures in various applications, such as cooking, chemistry, or industrial processes. In the given scenario, concentration allows us to track how the proportion of sugar changes as more sugar and water are added with time. By using the formula for concentration:

\[ C(t) = \frac{S(t)}{V(t)} \]where \( S(t) \) is the amount of sugar and \( V(t) \) is the volume of water, we determine the sugar content in the tank at any given moment \( t \). This rational function gives us a dynamic picture of how the concentration changes as more materials are added.

Modeling Change Over Time

Modeling change over time is crucial in understanding how quantities evolve in dynamic systems, such as our sugar-water mixture. In this exercise, we focus on how the amounts of sugar and water in the tank change with each passing minute. By modeling these changes, we can predict future conditions of the tank’s contents.

• Every minute, 3 pounds of sugar are added, captured by the function \( S(t) = 10 + 3t \).
• Simultaneously, 10 gallons of water are added, represented by \( V(t) = 200 + 10t \).

These linear functions describe how sugar and water accumulate as time progresses. Combining these separate processes in a single rational function helps in foreseeing how the system will behave after certain periods. Such modeling is not just restricted to mixtures but is widely applicable in fields like economics, population studies, and physics wherever change needs to be understood to aid decision-making.

Function Variables

Defining function variables is integral to translating real-world scenarios into mathematical models. In this task, two functions, \( S(t) \) and \( V(t) \), are used. Each represents a critical component of the situation: the sugar and the volume of water in the tank respectively.

• \( S(t) \) denotes the sugar content in pounds at time \( t \), starting with 10 pounds and increasing by 3 pounds each minute.
• \( V(t) \) indicates the volume of water in gallons, starting at 200 gallons and increasing by 10 gallons per minute.

Variables like \( S(t) \) and \( V(t) \) allow us to express changes quantitatively, leading to a functional description of the concentration over time. Function variables are essential tools for breaking down complex topics into manageable parts and are a foundational concept in calculus and algebra that students must grasp fully.