Problem 101 - College Algebra with Modeling and Visualization

Question

Working Together Suppose that a lawn can be raked by one gardener in 3 hours and by a second gardener in 5 hours. (a) Mentally estimate how long it will take the two gardeners to rake the lawn working together. (b) Solve part (a) symbolically.

Step-by-Step Solution

Answer

It will take about 1.875 hours for both gardeners to rake the lawn together.

Step 1: Estimate Joint Work Rate

First, let's estimate the time it will take for both gardeners working together to rake the lawn. If one can do it in 3 hours and the other in 5 hours, together it might take a bit less than the shortest individual time. Roughly, we can estimate around 2 hours.

Step 2: Define Work Rates Individually

The first gardener can complete the lawn in 3 hours, thus their work rate is \( \frac{1}{3} \) of the lawn per hour. The second gardener completes it in 5 hours, so their work rate is \( \frac{1}{5} \) of the lawn per hour.

Step 3: Calculate Combined Work Rate

Add the individual work rates to find the combined work rate:\[\text{Combined Work Rate} = \frac{1}{3} + \frac{1}{5}\]To add these, find a common denominator, which is 15:\[\frac{1}{3} = \frac{5}{15}, \quad \frac{1}{5} = \frac{3}{15}\]Thus, \(\frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15}\)},{

Step 4: Solve for Time Taken Together

Since the combined work rate is \( \frac{8}{15} \) of the lawn per hour, they will complete the lawn in:\[\text{Time} = \frac{1}{\left(\frac{8}{15}\right)} = \frac{15}{8} \text{ hours}\]Thus, it will take \( \frac{15}{8} \) hours, or 1 hour and 52.5 minutes, for both gardeners working together to finish raking the lawn.

Key Concepts

Joint Work

When two or more people collaborate on completing a task, the concept of joint work comes into play. This refers to combining their unique strengths or efficiencies to accomplish the task faster than working alone. In our example, two gardeners are tasked with raking a lawn. Alone, one can finish it in 3 hours and the other in 5 hours. By working together, they harness each other's abilities, speeding up the process. Joint work is commonly used in problems where collaborative effort leads to time-saving, showing the power of teamwork.

Estimating Work Time

Estimating work time is about making an educated guess on the time needed for a team to complete a task. Here, we want to find out how long it will take for both gardeners to finish raking the lawn together. A good estimation strategy for joint work problems involves considering the quickest individual’s time and estimating slightly less than that. If one gardener takes 3 hours, another 5 hours, a reasonable joint effort estimate is somewhere around 2 hours. Estimations like these help plan and manage time effectively even before computing the exact time mathematically.

Combined Work Rate

The combined work rate is crucial in determining how quickly a duo can complete a shared task. Each worker’s individual rate, expressed as units of work (like raking a lawn) per hour, are summed to ascertain how fast they can work together.

• First gardener’s rate: \( \frac{1}{3} \) of a lawn per hour
• Second gardener’s rate: \( \frac{1}{5} \) of a lawn per hour

To find the combined rate, add these together. Start by getting a common denominator, which is 15 in this case. Convert the rates: \( \frac{1}{3} = \frac{5}{15} \) and \( \frac{1}{5} = \frac{3}{15} \). Add them to find the combined rate: \( \frac{5}{15} + \frac{3}{15} = \frac{8}{15} \) lawns per hour. Knowing this combined rate allows you to calculate the joint time needed.

Symbolic Solution

Solving work rate problems symbolically involves using algebra to find precise solutions. In our lawn example, once we have the combined work rate, the next step is to determine the time it takes to finish the task. We found the joint work rate to be \( \frac{8}{15} \). Thus, the time \( T \) required is the reciprocal of this rate: \[T = \frac{1}{\left(\frac{8}{15}\right)}\]This simplifies to \( \frac{15}{8} \) hours. Converting to minutes gives us 1 hour and 52.5 minutes. By using a symbolic solution, we find exactness and clarity in how collaborative efforts translate to efficiency. This approach is particularly useful in various real-world applications, helping to optimize teamwork and plan effectively.