Problem 58 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Amir and Omar are tiling an area measuring \(A\) square meters. They lay down \(N\) tiles per square meter. Omar can put down \(q\) tiles in \(r\) hours while it takes Amir \(m\) minutes to lay one tile. \(A, N, q, r\), and \(m\) are constants. (a) Give the number of tiles Amir and Omar can put down as a function of \(t\), the number of hours they work together. (b) How many square meters can they tile in \(t\) hours? (c) After \(H\) hours of working with Omar, Amir leaves. The job is not yet done. How many hours will it take Omar to finish the job alone? Express the answer in terms of any or all of the constants \(A, N, q, r, m\), and \(H\).

Step-by-Step Solution

Answer

a) The number of tiles Amir and Omar can lay together in \(t\) hours is \(t * (q/r + 60/m)\) tiles.\n b) The total area they can tile in \(t\) hours is \(t * (q/r + 60/m) / N\) square meters.\n c) After Amir leaves, Omar alone will take \(((A * N) - (H * (q/r + 60/m)))/(q/r)\) hours to finish the job.

Step 1: Determine individual rates

Firstly, we need to determine how many tiles each individual can lay in an hour. Omar's rate is given by the ratio \(q/r\) tiles per hour. On the other hand, Amir's rate needs to be converted to tiles per hour. Since \(m\) is the time in minutes it takes for him to lay a tile, his rate is \(60/m\) tiles per hour.

Step 2: Determine joint rate

Next, we need to find out how many tiles both can lay together in an hour, which is simply the sum of their individual rates. Therefore, the rate for the two working together is \((q/r + 60/m)\) tiles per hour.

Step 3: Answer to Part (a)

Base on the result from the previous step, if they work together for \(t\) hours, the number of tiles they can lay together is therefore \(t * (q/r + 60/m)\) tiles.

Step 4: Answer to Part (b)

Given that each square meter requires \(N\) tiles, the total area they can tile together in \(t\) hours is the total number of tiles they can lay divided by the number of tiles per square meter. Therefore, the area they can tile in \(t\) hours is \(t * (q/r + 60/m) / N\) square meters.

Step 5: Answer to Part (c)

For the third part, after \(H\) hours of working with Omar, Amir leaves. Together they have tiled \(H * (q/r + 60/m)\) tiles. Suppose Omar alone takes \(X\) hours to finish the remaining, then we have \(H * (q/r + 60/m) + X * q/r = A * N\). Solving this equation for \(X\), we find \(X = ((A * N) - (H * (q/r + 60/m)))/(q/r)\), therefore, Omar will take \(((A * N) - (H * (q/r + 60/m)))/(q/r)\) hours to finish the job alone.

Key Concepts

Rate of Work

The rate of work is a measure of how much work is completed in a specific amount of time. For instance, in tiling problems, it refers to how many tiles can be laid per hour. This concept allows us to quantify work and predict the time needed to finish a job.
To determine the rate of work for Omar and Amir, we look at their individual contributions. Omar's rate is straightforward: he can lay down \(q\) tiles over \(r\) hours, equating to \(q/r\) tiles per hour. Amir's rate needs a bit of conversion. Since Amir lays one tile in \(m\) minutes, converting this to hours is essential. There are 60 minutes in an hour, so his rate becomes \(60/m\) tiles per hour.

Understanding each person's rate of work is crucial because it helps in planning and dividing tasks efficiently. It is a fundamental part of solving combined work problems and ensures that all parties are working to their fullest potential.

Collaboration in Work

Collaboration in work scenarios, like tiling, involves multiple workers combining their efforts to complete a task faster or more efficiently. Once we know the individual rates of work, we can find the combined rate when both individuals work together.

For Amir and Omar working together, their combined rate is simply the sum of their individual rates. This means they can lay down \(q/r + 60/m\) tiles per hour.

When calculating how much work they can finish together, we multiply this combined rate by the time they work (\(t\) hours). Their total production becomes \(t \times (q/r + 60/m)\) tiles after collaborating for \(t\) hours.

Collaboration in work illustrates how teamwork can efficiently solve problems that would take considerably longer individually. It highlights the importance of combining resources and skills to reach common goals effectively.

Algebraic Expressions

Algebraic expressions play a significant role in presenting solutions to word problems like the tiling task involving Amir and Omar. These expressions help us define relationships between various factors such as time, rate, and quantity of work.

Formulating these expressions requires understanding how the different variables interact. For instance, the number of tiles laid can be expressed as \(t \times (q/r + 60/m)\) when they work together for \(t\) hours, demonstrating how time and rate form the basis of algebraic relationships.

The power of algebraic expressions lies in their ability to simplify and solve complex problems by offering a systematic way to manipulate variables within equations. They transform practical scenarios, involving rates and collaborations, into manageable mathematical equations that provide clear and concise answers.

Unit Conversion

Unit conversion is crucial in solving tiling problems. It ensures a consistent measurement system, particularly when working with time. In the exercise, Amir’s rate of work was provided in minutes per tile, while Omar’s was in hours.

Before any calculation involving combined work, it’s necessary to convert Amir's rate to match Omar's, converting minutes into an hour-based rate. Since there are 60 minutes in an hour, Amir’s rate becomes \(60/m\) tiles per hour.

• This conversion allows for accurate comparison and summation with Omar’s \(q/r\) tiles per hour.
• Unit conversion ensures all rates are in a common system, facilitating straightforward calculations.

Unit conversion is an essential pre-step in calculations, ensuring that all elements of the problem work together harmoniously and providing clarity in multi-step mathematical solutions.