Problem 68 - Elementary Algebra

Question

If a cold water faucet, when opened, can fill a tank in 3 hours, how much of the tank is filled at the end of \(h\) hours? (See Figure 7.1.)

Step-by-Step Solution

Answer

\(\frac{h}{3}\) of the tank is filled after \(h\) hours.

Step 1: Determine the Faucet's Rate

To find out how much of the tank is filled in any given time, first determine the rate at which the faucet fills the tank. Since the faucet can fill the entire tank in 3 hours, its rate is \(\frac{1}{3}\) of the tank per hour.

Step 2: Calculate Tank Fill After h Hours

Using the rate from Step 1, calculate how much of the tank is filled after \(h\) hours. Multiply the rate by \(h\): \(\text{Tank filled} = \frac{1}{3} \times h = \frac{h}{3}\).

Step 3: Interpret the Result

Express the fraction \(\frac{h}{3}\) as a proportion of the tank. This fraction represents how much of the tank is filled after \(h\) hours based on the faucet's rate.

Key Concepts

Understanding Work Rate

In the world of rate problems, understanding work rate is crucial. A work rate is basically how fast a process occurs. In this exercise, it pertains to how quickly the cold water faucet can fill the tank with water. To simplify it, think of it as the amount of work completed in one unit of time. If the faucet fills the entire tank in 3 hours, its work rate is straightforwardly

• One-third of the tank per each hour

This fraction shows that every hour, the faucet completes filling \(rac{1}{3}\) of the tank. Consider it as a work cycle repeated hourly. Understanding this concept sets the foundation for solving more complex rate problems. It’s important to remember that work rate can be constant or variable, depending on the situation. Here, the faucet’s flow is constant, simplifying our calculations.

Using Fractions in Rate Problems

Fractions play a big role in solving rate problems. They provide a clear representation of parts of a whole. In our exercise, fractions help express how much of the tank is filled at any time. When the faucet fills \(rac{1}{3}\) of the tank in one hour, we establish a pattern. So, for each additional hour, we multiply \(rac{1}{3}\) by the number of hours passed, denoted as \(h\).

• This results in a simple expression: \(\frac{h}{3}\)

This fraction quickly tells us how much work has been done in terms of the tank being filled. Fractions are essential, as they provide a straightforward way to compare different rates or scales of work. With practice, you'll become more comfortable manipulating fractions to solve various kinds of rate problems.

Proportional Reasoning and Its Application

Proportional reasoning allows us to make sense of a situation using known relationships. In this exercise, it helps us stretch the use of fractions further. We use the faucet’s rate and time to determine how much of the tank is filled by understanding proportional relationships. Here are key steps for applying proportional reasoning:

• Identify the relationship between time and the part of the tank filled: \(\frac{1}{3}\) per hour.
• Understand that this rate is consistent over time. Therefore, formula \(\frac{h}{3}\) is proportional to the time \(h\).
• Apply the concept to predict or compare tank fill amounts by increasing or decreasing time.

This gives us the fraction \(\frac{h}{3}\) as an accurate indicator of how proportions of the tank fill depending on time. Proportional reasoning helps in extrapolating data from known values to unknown quantities, a skill quite useful in many practical problems beyond just faucets and tanks!