Problem 36 - Precalculus with Limits

Question

The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k}\right)\). After 20 days on the job, a new employee produces 19 units. Find the learning curve for this employee (first, find the value of \(k\) ). How many days should pass before this employee is producing 25 units per day?

Step-by-Step Solution

Answer

First, the learning rate or the value of \(k\) can be calculated by substituting into the expression \(k=\frac{1}{20}ln(1-\frac{19}{30})\). Then, substitute the value of \(k\) into the equation \(t=\frac{1}{k}ln(1-\frac{25}{30})\) to find the exact time period in which the worker will start producing 25 units a day.

Step 1: Find the value of k

We need to find the value of \(k\) using the model equation and the given data that after 20 days the worker produces 19 units. Substitute \(N=19\) and \(t=20\) into the equation to solve for \(k\): \(19=30(1-e^{20k})\). First, isolate the exponential term: \(e^{20k}=1-\frac{19}{30}\). Now, apply the natural logarithm to both sides to solve for \(k\): \(k=\frac{1}{20}ln(1-\frac{19}{30})\)

Step 2: Evaluate the value of k

Calculating the value of \(k\) from the previous step will give an approximate value for \(k\), which helps to understand how fast the worker's productivity increases.

Step 3: Use the value of k to find time

Now that we have the value of \(k\), substitute \(N=25\) in the model and solve for \(t\). Formulate the equation in this form: \(25=30(1-e^{kt})\). Isolate the exponential term: \(e^{kt}=1-\frac{25}{30}\). Apply the natural logarithm on both sides to solve for \(t\): \(t=\frac{1}{k}ln(1-\frac{25}{30})\). Calculate the approximate period when productivity reaches 25 units per day.

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which the variable appears in the exponent. It follows the general form of \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions are known for their ability to grow rapidly, which makes them very useful in modeling scenarios where change happens at an accelerating rate, such as population growth, interest rates, or, as in our case, learning curves in worker productivity.

• Exponential functions are continuous and smooth, meaning they don't have abrupt changes.
• They are often used to model processes that have a constant proportional growth rate.

In the context of the learning curve problem, the exponential function explains how the productivity of an employee increases over time. The variable \( t \) (days) is in the exponent, indicating that the rate at which productivity increases is proportional to the current level of productivity and not the mere passage of time.

Natural Logarithm

Natural logarithms are used to solve equations involving exponential functions. The natural logarithm, denoted as \( \ln(x) \), is the inverse function of the exponential function using base \( e \). This is particularly handy when you need to "bring down" an exponent to solve for a variable, like \( k \) or \( t \) in our learning curve problem.

• The natural logarithm of a number is the power to which \( e \) must be raised to obtain that number.
• It's key in solving equations where the unknown is in the exponent, as in \( e^{kt} \).

For instance, we use the natural logarithm to solve for \( k \) when determining how quickly the employee's productivity increases: \( k=\frac{1}{20}\ln(1-\frac{19}{30}) \). This equation helps interpret how rapidly productivity evolves over a set period of time.

Worker Productivity

Worker productivity in the context of this learning curve model refers to the number of units a worker can produce per day. It's an essential measure of efficiency and growth in any production setting. Understanding productivity helps in forecasting, resource allocation, and optimizing operations.

• Initially, productivity may start low as new employees learn tasks.
• Productivity increases over time as the employee becomes more skilled and efficient.
• The maximum potential productivity, set at 30 units in this model, represents the peak efficiency of a worker.

This model emphasizes the gradual improvement of a worker, highlighting the time-dependent nature of skill acquisition. As new employees spend more time working, their output increases, following a predictable pattern described by the exponential function.

Mathematical Modeling

Mathematical modeling involves using mathematical concepts and equations to represent real-world phenomena. It provides a framework for understanding complex systems by translating them into a form that can be analyzed and solved. In our problem, a learning curve model is applied to understand the productivity of workers over time.

• Helps to predict future behavior of a system based on past data.
• Enables testing of different "what-if" scenarios to plan for the future.
• In the given problem, the model \( N=30(1-e^{kt}) \) shows how productivity changes over time.

By using this model, we can calculate how quickly productivity increases, find the time needed to reach certain productivity goals, and determine optimal training periods. Such insights are crucial for efficient workforce management and planning.