Problem 61 - Elementary Algebra

Question

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. The water from a garden hose turned on at full pressure fills a hot tub in \(45 \mathrm{~min}\). If the drain is open, the hot tub empties in \(62 \mathrm{~min}\). Find the amount of time to fill the hot tub with the drain open. Round to the nearest whole number.

Step-by-Step Solution

Answer

164 minutes

Step 1 - Define the Rates

First, determine the rates. The hose fills the hot tub in 45 minutes, so the fill rate is \(\frac{1}{45}\) of the tub per minute. The drain empties it in 62 minutes, so the drain rate is \(\frac{1}{62}\) of the tub per minute.

Step 2 - Set Up the Equation

When the hose is filling the tub and the drain is open, the combined rate is \(\frac{1}{45} - \frac{1}{62}\) of the tub per minute. Let \(t\) be the time in minutes to fill the tub with both the hose filling and the drain open. The total amount filled in time \(t\) is 1 tub.

Step 3 - Simplify the Combined Rate

Calculate the combined rate as follows: \(\frac{1}{45} - \frac{1}{62}\). Find a common denominator to combine the fractions: \(\frac{62 - 45}{45 \times 62} = \frac{17}{2790}\). Hence, the combined rate is \[\frac{17}{2790}\] of the tub per minute.

Step 4 - Setting Up the Equation

The equation representing the situation is \[\left(\frac{17}{2790}\right)t = 1\]. This equation implies that the total rate multiplied by the time \(t\) equals one full tub.

Step 5 - Solve for Time

Solve for \(t\) by isolating it: \[t = \frac{2790}{17}\]. This simplifies to approximately \(t \approx 164\) minutes.

Key Concepts

work rate

Understanding the concept of work rate is essential for solving problems involving the time it takes to complete a task. In this context, work rate refers to the fraction of a task that is completed in a unit of time. For example, if a garden hose can fill a hot tub in 45 minutes, its work rate is \(\frac{1}{45}\) of the tub per minute. This rate remains constant whether the hose is filling the tub alone or with other factors involved, like a drain emptying the tub simultaneously. Essentially, the work rate tells you how fast a task is being completed over time.

combined work rate

When multiple factors influence the completion of a task, such as a hose filling and a drain emptying a hot tub at the same time, we need to calculate the combined work rate. The combined work rate is the net effect of all individual work rates involved. For example, the hose’s fill rate is \(\frac{1}{45}\) tubs per minute, and the drain's empty rate is \(\frac{1}{62}\) tubs per minute. Since the drain subtracts from the fill rate, the combined work rate becomes \(\frac{1}{45} - \frac{1}{62}\). This calculation gives us the net work rate, representing how quickly the tub is being filled when both the hose and drain are active. The combined work rate is essential for determining how much time it takes to fill the tub under these conditions.

algebraic equations

Solving problems like these often requires setting up and solving algebraic equations. Once you have your work rates, you can form an equation based on the relationship: \[ \text{combined work rate} \times \text{time} = \text{total work done} \] Here’s how it works in our example: Let \t\ be the time in minutes needed to fill the tub with the hose and drain operating together. The combined rate was found to be \(\frac{17}{2790}\) of the tub per minute. Therefore, the equation is \(\frac{17}{2790} \times t = 1\). This equation states that the combined rate multiplied by the time equals one full tub. Solving this equation will give us the value of \t\.

common denominators

When dealing with fractions in algebraic equations, finding a common denominator is crucial to simplifying and combining the fractions. For example, to combine \(\frac{1}{45}\) and \(\frac{1}{62}\), we need a common denominator. This common denominator is the least common multiple (LCM) of the two denominators, which in this case is \2790\. We then rewrite the fractions with this common denominator: \(\frac{62}{2790}\) and \(\frac{45}{2790}\). Subtracting these gives \(\frac{62 - 45}{2790} = \frac{17}{2790}\). Using common denominators helps streamline the process of solving for the variable and ensures our calculations are accurate.