Problem 24 - Precalculus: Functions and Graphs

Question

A swimming pool can be filled by three pipes, \(A, B,\) and \(C .\) Pipe \(A\) alone can fill the pool in 8 hours. If pipes \(A\) and \(C\) are used together, the pool can be filled in 6 hours; if \(B\) and \(C\) are used together, it takes 10 hours. How long does it take to fill the pool if all three pipes are used?

Step-by-Step Solution

Answer

The pool can be filled in 3 hours using all three pipes.

Step 1: Determine Pipe A's Rate

Pipe A fills the pool in 8 hours, so its rate is \( \frac{1}{8} \) of the pool per hour.

Step 2: Determine Combined Rate of Pipes A and C

Together, Pipes A and C fill the pool in 6 hours, so their combined rate is \( \frac{1}{6} \) of the pool per hour.

Step 3: Calculate Pipe C's Rate Using A and C Rates

The rate of Pipe C can be found by subtracting A's rate from the combined rate of A and C: \( \frac{1}{6} - \frac{1}{8} = \frac{1}{24} \). Pipe C's rate is \( \frac{1}{24} \) of the pool per hour.

Step 4: Determine Combined Rate of Pipes B and C

Together, Pipes B and C fill the pool in 10 hours, so their combined rate is \( \frac{1}{10} \) of the pool per hour.

Step 5: Calculate Pipe B's Rate Using B and C Rates

The rate of Pipe B can be found by subtracting C's rate from the combined rate of B and C: \( \frac{1}{10} - \frac{1}{24} = \frac{7}{120} \). Pipe B's rate is \( \frac{7}{120} \) of the pool per hour.

Step 6: Calculate Combined Rate of A, B, and C

Add up the rates of Pipes A, B, and C to get the rate when all three are used: \( \frac{1}{8} + \frac{7}{120} + \frac{1}{24} = \frac{15}{40} \) which simplifies to \( \frac{1}{3} \).

Step 7: Determine Time to Fill Pool with All Pipes

Since the combined rate of all pipes is \( \frac{1}{3} \) of the pool per hour, it takes 3 hours to fill the pool when all three pipes are in use.

Key Concepts

Rate of Work Problems

Rate of work problems involve determining how long it takes to complete a task when multiple entities contribute at different rates. In our scenario, three pipes are used to fill a swimming pool, each with a unique rate. Knowing how efficiently tasks can be completed requires understanding each entity's rate individually and collectively.

• Each task completes at its own pace, contributing to the whole task.
• Finding the total rate involves combining these individual rates.

By adding together the rates of all contributing entities, you can determine how long it will take to complete the task when all sources work simultaneously, providing a comprehensive solution to the problem.

Algebraic Fractions

Algebraic fractions play a crucial role in rate of work problems since rates are often expressed as fractions. These fractions represent the portion of the work done in a certain unit of time. For instance, if Pipe A fills the pool in 8 hours, its rate is represented as \( \frac{1}{8} \).

• Addition or subtraction of fractions is used when finding combined rates.
• Common denominators are needed to simplify these calculations.

To solve the pool problem, algebraic fractions are manipulated to find individual rates like at Step 3 in the original solution, where the rate of Pipe C is derived using fractions.

Systems of Equations

Systems of equations are used to solve for unknowns in problems with multiple variables. While not explicitly shown in the pool problem, the understanding of how rates interact reflects systems of equations principles.

• Each equation expresses the relationship between different rates and times.
• Unlike traditional systems of equations, these are solved through direct fraction manipulation.

By framing each rate equation like Steps 1 and 2 in terms of work done upon time, it's akin to solving a small system where the outcome is a combined rate solving for how quickly all pipes fill the pool.

Time and Work Problems

Time and work problems focus on how long it takes to complete a task with a certain number of participants working together or separately. In this exercise, the focus is on filling a pool using pipes A, B, and C.

• Understand individual work rates as portions of the task per unit time.
• Combine different work rates to determine total task time.

By adding up the individual rates of each pipe, as seen in Step 6, you identify how long it will take to complete the task when all are used simultaneously. Such problems teach combining individual contributions to find a total completion time.