Problem 11
Question
Each exercise is a problem involving work. You must leave for campus in 10 minutes or you will be late for class. Unfortunately, you are snowed in. You can shovel the driveway in 20 minutes and your brother claims he can do it in 15 minutes. If you shovel together, how long will it take to clear the driveway? Will this give you enough time before you have to leave?
Step-by-Step Solution
VerifiedKey Concepts
Algebraic Rates of Work
Understanding algebraic rates of work is crucial in solving problems that involve time and efficiency. This concept revolves around quantifying how fast a task is completed, which can be expressed with the formula: Rate = Work / Time. Let's dissect this with an example from an everyday situation: shoveling snow from a driveway.
Imagine you can clear your driveway in 20 minutes. Now, in algebraic terms, your rate of work is 1 driveway in 20 minutes, or mathematically represented as \(\frac{1}{20}\) driveways per minute. If you think of 'driveways cleared' as the unit of work, this fraction simply tells us how much work you can do in one minute, which is the essence of an algebraic rate.
- Always start by establishing the unit of work (e.g., one driveway cleared).
- Then, identify the time it takes to complete this work individually.
- Express the rate as a fraction with ‘work’ in the numerator and ‘time’ in the denominator.
Combining Work Rates
When you have multiple individuals or machines working on the same task, you need to understand combining work rates to calculate the overall efficiency. In our scenario, you're not shoveling snow alone; your brother joins in, with his rate of work being slightly faster. To find out how this teamwork affects the time taken, you combine the rates algebraically.
Adding your rate \(\frac{1}{20}\) driveways per minute to your brother's rate \(\frac{1}{15}\) results in a combined rate of \(\frac{1}{20} + \frac{1}{15} = \frac{3}{60} + \frac{4}{60} = \frac{7}{60}\) driveways per minute. It's important to find a common denominator when combining rates. This combined rate represents the new efficiency of the team working together — the more people working in tandem (with positive rates), the larger the combined rate and the less time the task will take.
- Calculate individual rates first before combining them.
- Find a common denominator to add rates effectively.
- Understand that the combined rate will always be higher than the individual rates, signifying increased efficiency.
Time Management in Problem Solving
Time management in problem solving is about making efficient use of available time — a skill that's just as critical in real-life situations as it is in mathematical problems. You have only 10 minutes before you need to leave for campus, and successfully managing this time is just as important as solving the problem itself.
In our example, once the combined work rate is established, calculating how long it takes to clear the snow becomes a simple division: divide 1 driveway by \(\frac{7}{60}\) driveways per minute to get approximately 7.5 minutes. This gives you an insight into whether the task can be completed within the constraints. Since 7.5 minutes is less than the 10 minutes available, you have enough time to shovel the snow and head to campus without being late.
- Always assess available time against the task duration.
- Efficient time management could involve combining efforts to meet deadlines.
- Concluding whether or not a task can be completed in the given time is crucial for decision-making.