Problem 58 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. All numbers are accurate to at least two significant digits. Two pipes together drain a wastewater-holding tank in 6.00 h. If used alone to empty the tank, one takes \(2.00 \mathrm{h}\) longer than the other. How long does each take to empty the tank if used alone?

Step-by-Step Solution

Answer

Pipe B takes approximately 11.1 hours, and Pipe A takes approximately 13.1 hours to empty the tank alone.

Step 1: Understanding the Problem

We have two pipes draining a tank together in 6 hours. If used separately, Pipe A takes 2 hours longer than Pipe B to drain the tank. We need to find out how many hours each pipe takes individually to empty the tank.

Step 2: Define Variables

Let \( x \) be the time (in hours) it takes for Pipe B to drain the tank alone. Consequently, Pipe A would take \( x + 2 \) hours to drain the tank alone.

Step 3: Express Rates of Work for Each Pipe

The rate of work for Pipe B is \( \frac{1}{x} \) (tank per hour) and for Pipe A is \( \frac{1}{x+2} \) (tank per hour).

Step 4: Setup the Equation for Combined Work

When both pipes work together, their combined rate is \( \frac{1}{6} \) (tank per hour). Hence, we can write the equation as \[ \frac{1}{x} + \frac{1}{x+2} = \frac{1}{6}. \]

Step 5: Solve the Equation

Multiply every term by \( 6x(x+2) \) to clear the denominators: \[ 6(x+2) + 6x = x(x+2). \] Expand and simplify: \[ 6x + 12 + 6x = x^2 + 2x. \] Combine like terms: \[ 12x + 12 = x^2 + 2x. \] Rearrange to form a quadratic equation: \[ x^2 - 10x - 12 = 0. \] Solve using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -10, c = -12 \).

Step 6: Calculate using the Quadratic Formula

Plug the values into the quadratic formula: \[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-12)}}{2(1)}. \] Simplify: \[ x = \frac{10 \pm \sqrt{100 + 48}}{2} = \frac{10 \pm \sqrt{148}}{2}. \] Calculate further: \[ x = \frac{10 \pm 12.17}{2}. \] Choose the positive root: \[ x = \frac{22.17}{2} \approx 11.085. \] Then, the other root is: \[ x = \frac{-2.17}{2} \approx -1.085. \] Since time can't be negative, \( x \approx 11.085 \).

Step 7: Calculate Time for Each Pipe

Since \( x \approx 11.085 \), Pipe B takes approximately 11.085 hours. Pipe A takes \( 11.085 + 2 = 13.085 \) hours.

Key Concepts

Rate of Work Problems

Rate of work problems are a fascinating area in mathematics that involve understanding how different agents, such as machines or people, can complete a task over time. In this context, let's think about two pipes emptying a tank, which is a practical example of rate of work. Each pipe can be thought of as an agent contributing to the task of draining a wastewater tank.
When dealing with rate of work problems, remember the concept of work rate, which is often expressed as a fraction of the task completed per time unit. For example, if a pipe can empty a tank in 6 hours, its rate is \( \frac{1}{6} \) tanks per hour.

• Define the individual rates: For two pipes working independently, identify how long each one takes individually to complete the task. In this case, suppose Pipe B takes \( x \) hours, giving a rate of \( \frac{1}{x} \) tanks per hour.
• Relate their rates: If Pipe A takes 2 hours longer than Pipe B, its rate becomes \( \frac{1}{x+2} \). The combined work rate when both pipes are operating is the sum of their individual rates.

Using these concepts, students can form equations that help solve for the unknowns, embodying a powerful strategy of equating total work to 1 (one whole task). This problem challenges the solver to think about how individual contributions combine towards a common goal.

Quadratic Equations

Quadratic equations are second-degree polynomial equations in the form of \( ax^2 + bx + c = 0 \). They appear frequently in algebra and are a fundamental concept in mathematics. In our tank problem, after setting up the work equation for the two pipes, we arrive at a quadratic equation that needs to be solved.
The equation obtained was \( x^2 - 10x - 12 = 0 \). Solving this requires the use of the quadratic formula:

• **Quadratic Formula:** \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• Here, the coefficients are: \( a = 1 \), \( b = -10 \), and \( c = -12 \).
• This formula helps find the values of \( x \) that satisfy the equation.

Substitute the coefficients into the formula and perform the calculations step by step. Through simplification, two potential solutions for \( x \) are found, but only the positive solution is meaningful in this context, as time cannot be negative. Thus, quadratic equations not only have roots but also bring logical analysis in real-life situation assessments.

Problem-Solving Strategies

Effective problem-solving in mathematics involves a series of strategic steps, especially when dealing with word problems. Here are some key strategies showcased in the given problem:

• **Understand the problem:** Start by reading the problem carefully. Identify what's being asked and what information is given. Break it down into manageable parts.
• **Define and set up variables:** Variables act as placeholders for the unknown quantities. Set these up systematically (e.g., \( x \) for the time Pipe B takes alone).
• **Formulate equations:** Use logical reasoning to translate the word problem into an equation or set of equations, based on the rates of work or relationships given.
• **Solve the equations:** Apply algebraic methods to find the solution. This may involve using well-established formulas, such as the quadratic formula, as seen here.
• **Interpret and verify the results:** Once a solution is found, check it against the original problem to ensure it makes sense and properly answers the question.

Throughout, it's crucial to stay organized and methodical, ensuring every step builds towards a viable solution. These strategies provide a framework for tackling a wide variety of mathematical challenges beyond this particular exercise.