Problem 68 - Calculus and its applications

Question

A company is producing a new product, and the time required to produce each unit decreases as workers gain experience. It is determined that $$T(x)=2+0.3\left(\frac{1}{x}\right)$$ where \(T(x)\) is the time, in hours, required to produce the \(x\) th unit. Use this information. Find the total time required for a worker to produce units 1 through 20 ; units 20 through 40 .

Step-by-Step Solution

Answer

The total time for units 1-20 is approximately 42.03 hours, and for units 20-40, it's approximately 39.92 hours.

Step 1: Understanding the Problem

The problem gives a function \(T(x) = 2 + 0.3\left(\frac{1}{x}\right)\) that represents the time in hours required to produce the \(x\)th unit. We need to find the total time to produce units 1 through 20 and units 20 through 40.

Step 2: Calculating Time for Units 1 through 20

The total time for units 1 through 20 is the sum \(T(1) + T(2) + \ldots + T(20)\). We calculate each \(T(x)\) and add them together.

Step 3: Calculating Time for Units 20 through 40

The total time for units 20 through 40 is the sum \(T(21) + T(22) + \ldots + T(40)\). Like before, calculate each \(T(x)\) and sum them.

Key Concepts

Learning Curve

The learning curve is a concept that describes how efficient workers become as they gain more experience and skills while performing a repetitive task. When a new task is introduced, workers often require more time to complete each unit. However, as they repeat the task, they learn better techniques and become more familiar, ultimately reducing the time needed per unit.

In the context of the given exercise, the learning curve is represented by the function \[ T(x) = 2 + 0.3\left(\frac{1}{x}\right). \]

• Here, \( T(x) \) indicates the time required to produce the \( x \)th unit.
• The formula shows that as \( x \) increases, the term \( 0.3\left(\frac{1}{x}\right) \), which depicts the learning aspect, decreases.

This reflects the idea that initially, the time decrease is more significant but reduces gradually as the worker gains experience.

Production Function

In economics and manufacturing, a production function describes the relationship between input resources and the resulting output. Essentially, it provides a mathematical way of describing how resources are transformed into products.

The function given in the problem, \[ T(x) = 2 + 0.3\left(\frac{1}{x}\right), \] demonstrates elements of a production function because it links the task of producing a unit (output) to the experience level, represented by the number \( x \), which can be considered an input.

• The constant term 2 represents a base time required for the production, regardless of experience levels.
• The term \( 0.3\left(\frac{1}{x}\right) \) indicates how the experience (input) impacts production time (output).

Such a function aids in strategically understanding and optimizing production processes by quantitatively capturing efficiency improvements with experience.

Summation in Calculus

Summation is a mathematical operation that involves adding a sequence of numbers, usually its function values across an interval. In calculus, it often deals with summing discrete values, as seen in the provided exercise.

To find the total time for producing a certain number of units, we sum the values of the function for each individual unit. The total time to produce units from 1 to 20 and from 20 to 40 is calculated by summing up:

• \( T(1) + T(2) + \ldots + T(20) \) for the first interval
• \( T(21) + T(22) + \ldots + T(40) \) for the second interval.

This approach of applying summation gives a precise total for the required production time across defined units. Moreover, it illustrates how calculus techniques can help simplify and solve real-world problems that involve accumulating effects, such as total working time or cumulative production cost over time.