Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem involving rate of work. A sink can be filled by the hot-water tap alone in 4 minutes more than it takes the cold-water tap alone. If both taps are open, it takes 4 minutes, 48 seconds to fill an empty sink. How long does it take each tap individually to fill the sink?

Step-by-Step Solution

Answer

Cold-water tap: 8 minutes; Hot-water tap: 12 minutes.

Step 1: Define Variables

Let \( x \) be the time (in minutes) it takes for the cold-water tap to fill the sink alone. Then, the hot-water tap takes \( x + 4 \) minutes to fill the sink alone.

Step 2: Set Up the Rates

The rate of the cold-water tap filling the sink is \( \frac{1}{x} \) sink per minute, and the rate of the hot-water tap is \( \frac{1}{x+4} \) sink per minute.

Step 3: Combine the Rates

When both taps are open, their combined rate is \( \frac{1}{x} + \frac{1}{x+4} \) sinks per minute. According to the problem, both taps together fill the sink in 4.8 minutes (since 4 minutes and 48 seconds is 4.8 minutes), which has the rate of \( \frac{1}{4.8} \) sinks per minute.

Step 4: Set Up the Equation

We can set up the equation for the combined work rate: \[ \frac{1}{x} + \frac{1}{x+4} = \frac{1}{4.8} \]

Step 5: Combine Fractions

Add the fractions on the left side of the equation: \[ \frac{(x+4) + x}{x(x+4)} = \frac{1}{4.8} \] This simplifies to \[ \frac{2x + 4}{x(x+4)} = \frac{1}{4.8} \]

Step 6: Cross Multiply

Cross multiply to get a solvable equation: \[ 2x + 4 = \frac{x(x+4)}{4.8} \] Multiply both sides by 4.8 to clear the fraction: \[ 4.8(2x + 4) = x^2 + 4x \]

Step 7: Expand and Simplify

Expand and collect like terms: \[ 9.6x + 19.2 = x^2 + 4x \] Rearrange to form a quadratic equation: \[ x^2 - 5.6x - 19.2 = 0 \]

Step 8: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -5.6, c = -19.2 \). Calculate the discriminant: \[ (-5.6)^2 - 4(1)(-19.2) = 31.36 + 76.8 = 108.16 \] Now solve for \( x \): \[ x = \frac{5.6 \pm \sqrt{108.16}}{2} \] Calculate \( \sqrt{108.16} = 10.4 \). Thus, \[ x = \frac{5.6 \pm 10.4}{2} \] Choose the positive solution \( x = 8 \) minutes.

Step 9: Determine Each Tap's Time

Since \( x = 8 \) minutes for the cold water tap, the hot water tap takes \( x + 4 = 12 \) minutes.

Key Concepts

Algebraic Equations

Algebraic equations are vital tools for representing real-world problems with mathematical expressions. In the context of rate of work problems, such as the one involving taps filling a sink, algebraic equations are used to model the relationships between different rates and times. An algebraic equation is a mathematical statement of equality that can contain variables, numbers, and operation symbols. In this exercise, we defined variables to represent the unknowns:

• Letting \( x \) be the time it takes for the cold-water tap to fill the sink.
• Expressing the hot-water tap’s time as \( x + 4 \) minutes.

By setting up an equation based on the rates of work, we can solve for \( x \). Remember, the key is understanding how different parts of the problem relate to form a complete picture. In this case, it's how the individual rates add together to equal the combined rate when both taps are operating concurrently.

Quadratic Formula

The quadratic formula is a critical tool for solving equations of the form \( ax^2 + bx + c = 0 \). It allows us to find the values of \( x \) that satisfy the equation by calculating:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In our exercise, after setting up and rearranging our equation, we ended up with a quadratic equation: \( x^2 - 5.6x - 19.2 = 0 \). The coefficients here are \( a = 1 \), \( b = -5.6 \), and \( c = -19.2 \). We use these values in the quadratic formula, calculating the discriminant \( b^2 - 4ac \), which tells us about the nature of the roots. A positive discriminant allowed us to find two distinct real solutions, from which we selected the positive one to make sense in our context and solve the problem meaningfully.

Problem Solving Steps

Breaking down a problem into manageable steps is crucial for finding a solution effectively. Here's an approach that works well for rate of work problems:
1. **Understand the Problem**: Identify what you know and what you need to find. 2. **Define Variables**: Assign symbols to unknown quantities that need to be determined. 3. **Formulate Equations**: Use the relationships given in the problem to form algebraic equations. 4. **Solve the Equations**: Choose appropriate methods to solve the equations you have set up, such as substitution or applying the quadratic formula.
In our example, following these steps helped us systematically approach the rate problem involving filling a sink with two taps. Each step builds on the previous one, ensuring that when you solve, you have a logical and complete solution path.

Basic Algebra Concepts

Understanding basic algebra concepts lays the foundation for solving complex problems. This includes knowing how to manipulate algebraic expressions, perform operations with fractions, and solve equations.

• **Variables and Expressions**: Understand what a variable is and how it stands in for unknown numbers in an equation.
• **Combining Like Terms**: Adding or subtracting terms that have the same variables and powers.
• **Operations with Fractions**: Knowing how to add, subtract, multiply, and divide fractions is crucial. Remember the common denominators when adding fractions and simplification of expressions thereafter.
• **Rearranging Equations**: Skills in moving terms from one side of an equation to the other are key in isolating variables.

Mastering these concepts allows you to tackle algebra problems confidently and prepares you for more advanced mathematics. By applying these basics, you'll be able to engage with even challenging exercises like the one involving rate of work with ease.