Problem 4 - Technical Mathematics with Calculus

Question

A technician can assemble an instrument in 9.5 h. After working for \(2.0 \mathrm{h},\) she is joined by another technician who, alone, could do the job in 7.5 h. How many additional hours are needed to finish the job?

Step-by-Step Solution

Answer

Additional hours needed to finish the job is approximately 2.68 hours.

Step 1: Determine the work rate of each technician

Let the work done by the first technician be represented by the fraction \(\frac{1}{9.5}\) of the job per hour and the second technician's work be \(\frac{1}{7.5}\) per hour.

Step 2: Calculate the amount of work completed by the first technician in 2 hours

The first technician works alone for 2 hours, so the work completed is \(2 \times \frac{1}{9.5} = \frac{2}{9.5}\) of the entire job.

Step 3: Calculate the combined work rate of both technicians working together

When both technicians work together, they can complete \(\frac{1}{9.5} + \frac{1}{7.5}\) of the job per hour. Combine the fractions to find a common work rate per hour.

Step 4: Determine the remaining work to be done

Subtract the work already done (\(\frac{2}{9.5}\)) from the entire job (represented by 1) to find the remaining fraction of the job: \(1 - \frac{2}{9.5}\).

Step 5: Calculate the time needed to complete the remaining job

Divide the remaining fraction of work by the combined hourly work rate of both technicians to find the number of additional hours needed to finish the job.

Key Concepts

Fractional Work Rates

Understanding fractional work rates is essential in solving problems related to individual contributions to a task over time. Imagine a pie representing the whole task. A fractional work rate like \(\frac{1}{9.5}\) implies that every hour, a technician completes 1 slice out of the 9.5 slices that make up the entire pie. Similarly, \(\frac{1}{7.5}\) work rate indicates the second technician would complete one out of 7.5 slices per hour.

When we talk about one technician joining another, we essentially 'add' their slices together to understand how big of the pie they can handle in one hour. This is why calculating the sum of two fractional work rates allows us to understand the combined efforts of both technicians.

Technician Work Rate

To calculate an individual's work rate, we take the inverse of the time it takes for them to complete a task alone. For our technician who can complete the job in 9.5 hours, her work rate is 1 divided by 9.5, or \(\frac{1}{9.5}\) of the job per hour.

This rate is crucial for planning and understanding how much work can be accomplished over a certain period. It is also a fundamental value we use to determine how additional resources (like a second technician) can accelerate the overall work completion.

Work Rate Calculation

To calculate the work rate, we must combine the individual rates of both technicians. For the first 2 hours, the first technician works alone at her rate of \(\frac{1}{9.5}\). This is where we calculate the fractional work completed, as mentioned earlier. Now, when the second technician joins in, the combined work rate becomes \(\frac{1}{9.5} + \frac{1}{7.5}\).

To find this combined rate, we look for a common denominator and add the fractions. This combined work rate then tells us how efficient the technicians are when working together, which is key to determining how quickly they can finish the remaining task.

Job Completion Time

To find out the job completion time after the first technician has already worked for 2 hours alone, we first calculate the amount of work left. We then divide this remaining work by the combined work rate of both technicians. The formula is essentially remaining job divided by work rate equals time needed.

The job completion time tells us how long it will take for the technicians to finish the rest of the work once they team up. This step is integral as it translates work rate into a tangible duration and helps to estimate project timelines effectively, ensuring timely job completion.