Question

You are hired to wash the new cars at a car dealership with two other employees. You take an average of 40 minutes to wash a car \(\left(R_1=1 / 40\right.\) car per minute \()\). The second employee washes a car in \(x\) minutes. The third employee washes a car in \(x+10\) minutes. a. Write expressions for the rates that each employee can wash a car. b. Write a single expression \(R\) for the combined rate of cars washed per minute by the group. c. Evaluate your expression in part (b) when the second employee washes a car in 35 minutes. How many cars per hour does this represent? Explain your reasoning.

Short answer

After substituting \(x = 35\) into our expression for \(R\) and calculated, we get R in per minute. Multiplied with 60, we can get the combined rate per hour, denoted as \(R_h\). This represents the number of cars the group can wash in an hour.

Step-by-step solution

Expressing Rates Individually

We usually express the rate at which work is done as 'work per unit time'. Here, the unit of time is minutes. Hence, the rate at which the first employee washes cars is \(R_1 = \frac{1}{40}\) cars per minute. The second employee's rate, expressed as cars washed per minute, is \(R_2 = \frac{1}{x}\). The third employee's rate is \(R_3 = \frac{1}{x+10}\).

Calculating Combined Rate

The combined rate of washes car by the group per minute is simply the sum of the individual rates: \(R = R_1 + R_2 + R_3 = \frac{1}{40} + \frac{1}{x} + \frac{1}{x+10}\).

Evaluating at a Specific Value of x

Given that the second employee takes 35 minutes to wash a car, we can substitute \(x = 35\) into our expression for \(R\): \(R = \frac{1}{40} + \frac{1}{35} + \frac{1}{45}\). We can convert these fractions to decimals to evaluate the expression for easier calculation.

Conversion to per hour

To convert the rate from per minute to per hour we simply multiply with 60. Hence, the combined rate per hour, \(R_h = R \times 60\). Calculate this value using the decimal form gotten from previous step.

Key concepts

Understanding Work Rate Expressions

When dealing with problems that include tasks being completed by different individuals or machines, we often use work rate expressions to describe how quickly work is done. This is usually expressed as the amount of work done per unit of time. In our scenario, the work is washing cars, and time is measured in minutes.

For instance, if it takes you 40 minutes to wash a car, your work rate is described as the number of cars you can wash per minute. Mathematically, this is represented as a fraction, with the number of cars on the numerator and the time in minutes on the denominator, which is \( R_1 = \frac{1}{40} \) cars per minute.

Similarly, if we don't know the exact time it takes for the second employee to wash a car, we represent it as \( R_2 = \frac{1}{x} \) where \( x \) is the number of minutes. For the third employee, who takes 10 minutes longer than the second, the rate is \( R_3 = \frac{1}{x+10} \).

Remember:

• The work rate expression is a way of quantifying efficiency.
• Work rate problems often assume a consistent work pace over time.
• Unit conversions may be necessary, but work rates will typically be consistent across these conversions.

By establishing a clear and consistent work rate expressions, we can solve problems involving multiple workers with different efficiencies and find a combined work rate for any group of workers.

Combining Work Rates

When multiple individuals or factors contribute to finishing a job, we talk about combined work rates. Understanding how to combine these rates is crucial for teamwork efficiency problems, as it gives us the total work rate of a group.

To find the combined work rate, you simply add up the individual rates, assuming that they are working simultaneously and independently. This will tell you how much work can be done per unit time by the entire group working together.

In the car washing example, the combined work rate for all three employees is \( R = R_1 + R_2 + R_3 \). By replacing each individual's rate, we obtain the expression \( R = \frac{1}{40} + \frac{1}{x} + \frac{1}{x + 10} \) cars per minute.

Important to Note:

• The combined rate is always the sum of the individual rates when working simultaneously.
• It is necessary to ensure that all rates are in the same unit before combining them.
• Combined work rates do not necessarily scale linearly with more workers due to potential diminishing returns or other factors.

This concept also applies to machines, where each machine has its own work rate, and the combined output rate is of interest when the machines are working at the same time.

Evaluating Expressions for Specific Conditions

The process of evaluating expressions involves substituting numbers for variables in an algebraic expression and then simplifying to get a numerical result. It is a fundamental part of solving work rate problems when you need to identify a specific scenario. For example, if you know the time it takes for each employee to wash a car, you can evaluate the combined rate.

In the case that the second employee takes 35 minutes to wash a car, you insert this value into the combined work rate expression: \( R = \frac{1}{40} + \frac{1}{35} + \frac{1}{45} \). After substitution, the next step is to simplify or calculate this combined rate. Transforming the fractions into decimals can make this step easier and is often necessary for real-world applications.

To get the total rate in terms of cars per hour, you would multiply the combined rate per minute by 60 since there are 60 minutes in an hour: \( R_h = R \times 60 \).

When evaluating expressions:

• Ensure that you substitute the correct values for each variable.
• Pay attention to the units being used, especially if you need to convert units, like from minutes to hours.
• Always simplify the expression as much as possible for straightforward interpretation.

Ultimately, evaluating the expressions in relation to work rate problems can determine efficiency and productivity rates for various work scenarios.