Problem 59 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sam Wright plays the role of Sebastian the crab in the Disney film "The Little Mermaid." He spent \(H\) hours working on the production. \(P\) percent of this time was spent on the taping; of the remaining time, \(1 / n\) was spent on rehearsal and the rest on dubbing and looping. Sam was paid \(D\) dollars per hour for each of the \(H\) hours spent on the production. (a) If Disney had changed the contract so that they paid for taping, dubbing, and looping but not rehearsal, how much would Sam's pay have been? (b) If Disney were paying for taping only, and Sam wanted to earn the same thing he would have under the original contract, how much would he need to charge per hour?

Step-by-Step Solution

Answer

(a) The pay would have been \(0.01PDH + (1-1/n)D(1-0.01P)H\). (b) The hourly rate would need to be \(D(1 + 0.01P)\) if only taping is paid in order for Sam to earn the same amount as the original contract.

Step 1: Calculate Hours spent and Pay for each Activity

Sam spent \(P\) percent of his total time taping, so the hours he spent taping can be represented as \(0.01P * H\). The remaining time is \((1-0.01P)H\), out of this, \(1/n\) is spent on rehearsal and the rest on dubbing and looping. This will give us the hours spent on dubbing and looping as \((1-1/n)(1-0.01P)H\). It's given that Sam is paid \(D\) dollars per hour. So, for each activity, the pay would be \(0.01PDH\) for taping, \((1-1/n)D(1-0.01P)H\) for dubbing and looping and \((1/n)D(1-0.01P)H\) for rehearsal.

Step 2: Calculate the Pay if Only Taping, Dubbing, and Looping are Paid

In scenario (a), if only taping, dubbing, and looping are paid, we should add the pay of taping and dubbing & looping hours and discard rehearsal pay. So the total pay in this case would be \(0.01PDH + (1-1/n)D(1-0.01P)H\).

Step 3: Calculate the Hourly Rate if Only Taping is Paid

In scenario (b), Sam wants his total pay to remain the same as in the original scenario but only taping is paid. Hence, the new hourly rate needed can be obtained by dividing his original total pay by the original taping hours. That would be \(\frac{(1-0.01P)DH + 0.01PDH}{0.01PH} = D(1 + 0.01P)\).

Key Concepts

Percentage calculations

Understanding percentage calculations is crucial, especially when dealing with scenarios involving allocation of resources or time, such as in Sam's contract. To determine the amount of time Sam spent on different activities, we start by calculating the percentage of hours dedicated to each task.

Here's how you do it: If Sam worked for a total of \(H\) hours, and \(P\) percent of this time was spent on taping, you convert this percentage into a decimal by dividing by 100. Thus, the calculation for the hours spent taping becomes \(0.01 \times P \times H\).

This technique helps in breaking down larger totals into understandable chunks. It's a handy tool not just for contract work, but for budgeting, time management, and even in personal finance. Once you grasp this concept, any task where a whole is divided into parts becomes much simpler.

Time management problems

Time management in Sam's scenario involves distributing his total hours among different production activities: taping, rehearsal, and dubbing/looping. Understanding the distribution of time helps to prioritize or rearrange tasks based on importance or effect on pay.

Let's evaluate: After finding out what fraction \((1/n)\) of the remaining hours was used for rehearsal, you can determine how much time is left for other activities through the formula: \((1 - \frac{1}{n})(1 - 0.01P)H\).

This shows the importance of efficient time allocation, particularly in environments where certain activities might contribute more significantly to overall objectives or financial outcomes. Mastering these skills can lead to more effective workflows and increased productivity, crucial in professional and educational settings alike.

Rate of change in contracts

Changes in contract specifics, such as the ones Sam faces, reflect how pay can be affected by shifting focus from multiple activities to a single one like taping. Understanding this concept requires considering how a change in one variable, like the hourly pay rate for taping, must compensate for the lack of pay in other areas.

By deriving the equation \(D(1 + 0.01P)\) for the new hourly rate, we ensure Sam's total pay remains steady even when income streams change. This involves calculating his total pay in the original scenario and dividing it by the new conditions, here focusing solely on taping.

Grasping this rate of change can illuminate how adjustments might be necessary in your own circumstances, be it adjusting budgets when priorities shift or negotiating contracts. It emphasizes the flexibility and strategic foresight needed in managing financial changes effectively.