- Define the work formula: \(\text{Work} = \text{Rate} \times \text{Time}\).
- Identify Dominik's rate as \(\frac{1}{8}\) part/hour and time as \(5\) hours.
- Calculate the work: \(\frac{1}{8} \times 5 = \frac{1}{8}(5)\).
- Final answer: \(\boxed{\frac{1}{8}(5)}\).
Explanation
- Analysis of the Work Problem
The problem asks us to determine the missing value in a table representing the portion of a project completed by Dominik, who works at a rate of \(\frac{1}{8}\) part/hour for 5 hours. The fundamental relationship in work-related problems is defined by the formula \(\text{Work} = \text{Rate} \times \text{Time}\). In this context, the rate of work is given as \(\frac{1}{8}\) part per hour, and the duration Dominik works is 5 hours. Our objective is to calculate the work completed by Dominik using this information.
- Calculating the Completed Work
To find the missing value, we apply the work formula. Dominik's rate is \(R_D = \frac{1}{8}\) part/hour and the time he spends working is \(T = 5\) hours. Therefore, the part of the project completed by Dominik is given by the product:
\[\text{Part of Project Completed} = R_D \times T = \frac{1}{8} \times 5 = \frac{1}{8}(5)\]
This calculation represents the portion of the task Dominik finishes within the 5-hour period.
- Selecting the Correct Option
Comparing our result \(\frac{1}{8}(5)\) with the provided options, we can clearly see that it matches the fourth option. The other options, such as \(8r\), \(8(5)\), and \(\frac{1}{8}r\), do not correctly represent the multiplication of Dominik's rate and his time. Thus, the missing value is \(\frac{1}{8}(5)\).
Examples
Consider a factory worker assembling parts for a machine. If the worker can assemble \(\frac{1}{20}\) of a machine per hour and they work for 6 hours, the total portion of the machine assembled would be \(\frac{1}{20} \times 6 = \frac{6}{20}\) or 30% of the machine. Understanding this relationship helps in project management, planning production schedules, and estimating the time required to complete various tasks effectively.