Question

You are running water into a laundry sink to get a mixture that is one-half hot water and one-half cold water. The hot water flows more slowly, at a rate of 7.8 liters per minute, so you turn it on first. Two minutes later, you also turn on the cold water which flows at a rate of 12.3 liters per minute. You want to know how long to wait before turning the two faucets off. a. Let \(t\) represent the number of minutes the hot water is on. Write a variable expression for the amount of time the cold water is on. b. Write an equation that models the situation. Then solve the equation. c. Writing Interpret your solution. If the solution is a decimal, decide what form of the number is most appropriate. Explain your choice.

Short answer

Hot water should be allowed to flow for approximately 5 to 6 minutes to ensure a mixture of half hot and half cold water in the sink.

Step-by-step solution

Define Variables

Let's denote \(t\) to be the total number of minutes the hot water is flowing. Considering the fact that the cold water starts flowing 2 minutes later, the cold water must be flowing for \(t-2\) minutes.

Establish the Equation

The problem can be represented as an equation by considering the fact that the total quantity of hot and cold water needs to be equal for the mixture to be one-half hot and one-half cold. Let’s denote the quantity of hot water as \(H\) and cold water as \(C\). Thus we get a following equation: \(7.8t=12.3(t-2)\)

Solve the Equation

To solve for \(t\), we first distribute the 12.3 to both terms inside the parentheses on the right-hand side. That results in the equation: \(7.8t = 12.3t - 24.6\). Next we rearrange the equation to get all terms involving \(t\) on one side by subtracting \(12.3t\) from both sides: \(-4.5t = -24.6\). Finally, we divide both sides by -4.5 to solve for \(t\): t = \(\frac{-24.6}{-4.5} = 5.47\)

Interpret the Solution

The solution to the equation is a little over 5 and a half minutes (5.47 minutes). But since exact timing to the decimals might not be feasible in a real-world scenario, we can approximate this to the nearest minute. Thus, you might want to let the hot water flow for about 5 to 6 minutes.

Key concepts

Systems of Linear Equations

Understanding the concept of systems of linear equations is integral to solving many real-world problems. In the exercise provided, we are faced with an equation that represents the balance between the amount of hot and cold water in a sink. If we imagine that there is another condition to be met, like a total volume constraint or a time constraint, we would have a system of linear equations.
When dealing with systems, we aim to find a common solution to multiple equations, just as in the exercise, we found a single time duration that satisfied the equation for equal hot and cold water.

• A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable.
• A system of these equations can be solved using various methods including graphing, substitution, and elimination.
• In real-world scenarios, these systems can model different constraints and conditions need to be satisfied simultaneously.

Not every system has a unique solution; some have no solution or an infinite number of solutions. Nonetheless, they are powerful tools for modeling and solving complex situations.

Rate of Change

The concept of rate of change is frequently encountered in algebra and represents how one quantity changes in relation to another. The exercise presents a practical instance where the rate of flow of hot and cold water determines how we approach the problem.
Here, the rate of change is the amount of water (in liters) flowing per minute for both hot and cold water faucets. Understanding and calculating the rate of change are essential for setting up the correct equations to solve real-world problems.

• Rate of change is a measure of how a quantity changes over time.
• It is often represented as a ratio or a fraction; in this scenario, as liters per minute.
• The concept ties closely with slope in graphing linear equations, where slope represents the rate of change.

In practical terms, being able to calculate the rate of change helps in predicting future outcomes and making informed decisions based on the current information.

Solving Algebraic Equations

Solving algebraic equations is a foundational skill in mathematics. In the exercise, we see a demonstration of how to approach and solve an algebraic equation step by step. It highlights the importance of clear and logical reasoning in finding the value of the variable.
Each step taken to solve the equation is methodical:

• Step 1: Define the variables representing real-world quantities.
• Step 2: Form an equation based on the conditions given.
• Step 3: Manipulate the equation using algebraic rules to isolate the variable.

It is crucial to be comfortable with handling variables, constants, and the arithmetic of combining like terms. The process often involves making the equation simpler or transforming it into a more solvable form, such by eliminating fractions or collecting like terms.
The ability to solve algebraic equations is not only critical for academic success in mathematics but also for solving complex problems in sciences, engineering, economics, and many other fields where mathematical modeling is employed.