Problem 38 - Beginning and Intermediate Algebra

Question

Solve each problem. Round answers to the nearest tenth as needed. Two pipes together can fill a tank in \(2 \mathrm{hr}\). One of the pipes, used alone, takes \(3 \mathrm{hr}\) longer than the other to fill the tank. How long would each pipe take to fill the tank alone?

Step-by-Step Solution

Answer

The faster pipe takes 3 hours, and the slower pipe takes 6 hours.

Step 1 - Define Variables

Let the time taken by the faster pipe to fill the tank alone be x hours. Hence, the slower pipe will take (x + 3) hours to fill the tank alone.

Step 2 - Set Up Equations for Rates

The rate of the faster pipe is \(\frac{1}{x}\) tanks per hour. The rate of the slower pipe is \(\frac{1}{x+3}\) tanks per hour. Together they fill the tank in 2 hours, so their combined rate is \(\frac{1}{2}\) tanks per hour.

Step 3 - Write the Combined Rate Equation

Set up the equation for their combined rates: \[ \frac{1}{x} + \frac{1}{x+3} = \frac{1}{2} \]

Step 4 - Clear the Denominators by Multiplying

Multiply through by \(2x(x+3)\) to clear the denominators: \[ 2(x+3) + 2x = x(x+3) \]

Step 5 - Simplify the Equation

Simplify to get: \[ 2x + 6 + 2x = x^2 + 3x \] Combine like terms: \[ 4x + 6 = x^2 + 3x \] Rearrange into standard quadratic form: \[ x^2 - x - 6 = 0 \]

Step 6 - Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve \(x^2 - x - 6 = 0\). Here, \(a = 1\), \(b = -1\), and \(c = -6\). Substitute these values in: \(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-6)}}{2(1)} \) which simplifies to \(x = \frac{1 \pm \sqrt{1 + 24}}{2} \) and further to \( x = \frac{1 \pm 5}{2} \). Thus, we get \(x = 3\) and \(x = -2\).

Step 7 - Choose the Positive Solution

Since time cannot be negative, the solution is \(x = 3\). Therefore, the faster pipe takes 3 hours and the slower pipe takes \(3 + 3 = 6\) hours.

Step 8 - Verify the Solution

Verify by checking their combined rate: \( \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \), confirming that they fill the tank together in 2 hours.

Key Concepts

Quadratic Equations

Understanding quadratic equations is essential for tackling algebra word problems like the one about filling a tank with two pipes. Quadratic equations are polynomial equations in the form \text{ax}^2 + \text{bx} + \text{c} = 0. To solve them, we often use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula helps us find the values of \(x\) that satisfy the quadratic equation. In the problem at hand, we formed a quadratic equation to determine the time taken by each pipe to fill the tank. We simplified the equation to \(x^2 - x - 6 = 0\), and by substituting the values \(a = 1\), \(b = -1\), and \(c = -6\) into the quadratic formula, we solved for \(x\) to get \(x = 3\) and \(x = -2\). However, since time cannot be negative, we chose \(x = 3\), meaning the faster pipe takes 3 hours to fill the tank alone.

Rate Problems

Rate problems involve calculating the speed or efficiency at which something is done. In the context of the tank-filling problem, we are dealing with the rates at which two pipes can fill the tank. If a pipe can complete a task in \(x\) hours, its rate is the reciprocal: \(\frac{1}{x}\) tanks per hour. Similarly, if another pipe takes \(x+3\) hours, its rate is \(\frac{1}{x+3}\). By combining these rates, we formed an equation: \( \frac{1}{x} + \frac{1}{x+3} = \frac{1}{2} \). This shows how the rates of both pipes add up to complete the task in 2 hours. Solving rate problems often involves setting up such equations, clearing denominators, and simplifying to find the unknown variables.

Solving Systems of Equations

Solving systems of equations is a method used to find values of variables that satisfy multiple equations simultaneously. In our exercise, we used a system of algebraic equations to find the time each pipe takes to fill the tank. We started by defining our variables, where \(x\) is the time taken by the faster pipe. Next, we created a system of rate equations. By solving the combined rate equation \( \frac{1}{x} + \frac{1}{x+3} = \frac{1}{2} \), we cleared the denominators and simplified the expression to form a quadratic equation \(x^2 - x - 6 = 0\). We then solved this system using the quadratic formula. Solving systems of equations often involves substitution and elimination methods, which help in finding the correct values of variables in complex problems.