Problem 67 - Calculus and its applications

Question

IA 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 for Exercises 67 and 68. Find the total time required for a worker to produce units 1 through \(10 ;\) units 20 through \(30 .\)

Step-by-Step Solution

Answer

Units 1-10: 20.978 hours, Units 20-30: 22.137 hours.

Step 1: Understand the Problem Statement

We have a time function \( T(x) = 2 + 0.3\left(\frac{1}{x}\right) \) that gives the time to produce the \( x \)-th unit. We need to compute the sum of production times over specified unit ranges: units 1 through 10, and units 20 through 30.

Step 2: Calculate Total Time for Units 1 to 10

Compute the time for each unit from 1 to 10 using the function and sum these values:\[\text{Total Time for Units 1 to 10} = \sum_{x=1}^{10} T(x) = \sum_{x=1}^{10} \left( 2 + 0.3 \frac{1}{x} \right)\]Calculate each term and add them up:- \( T(1) = 2 + 0.3 \cdot 1 = 2.3 \)- \( T(2) = 2 + 0.3 \cdot \frac{1}{2} = 2.15 \)- Continue similarly for \( x = 3 \text{ to } 10 \).- Sum: \( 2.3 + 2.15 + 2.1 + 2.075 + 2.06 + 2.05 + 2.042857 + 2.0375 + 2.033333 + 2.03 = 20.978690 \) hours approximately.

Step 3: Calculate Total Time for Units 20 to 30

Compute the time for each unit from 20 to 30 using the function and sum these values:\[\text{Total Time for Units 20 to 30} = \sum_{x=20}^{30} T(x) = \sum_{x=20}^{30} \left( 2 + 0.3 \frac{1}{x} \right)\]Calculate each term:- \( T(20) = 2 + 0.3 \cdot \frac{1}{20} = 2.015 \)- \( T(21) = 2 + 0.3 \cdot \frac{1}{21} = 2.014286 \)- Continue similarly for \( x = 22 \text{ to } 30 \).- Sum: \( 2.015 + 2.014286 + 2.013636 + 2.013043 + 2.0125 + 2.012 + 2.011538 + 2.01111 + 2.010714 + 2.010345 + 2.01 = 22.137172 \) hours approximately.

Key Concepts

Understanding Time Function in Calculus

In calculus, a time function is a mathematical representation that describes how time varies with respect to another variable, often related to practical scenarios such as production. Here, the time function provided, \( T(x) = 2 + 0.3 \left(\frac{1}{x}\right) \), tells us about the time needed in hours to produce the \( x \)-th unit of a product. This function helps us understand diminishing time as experience or the unit count increases.

Let's break down this function:

• The constant 2 represents a base time required for production regardless of experience.
• The term \( 0.3 \left(\frac{1}{x}\right) \) represents a decreasing addition to the base time as \( x \) increases, reflecting learning or efficiency improvements.

As \( x \) becomes larger, \( \frac{1}{x} \) gets smaller and so does the value of the \( 0.3 \left(\frac{1}{x}\right) \) part, hence the whole function reduces over unit production. This model is a great example of how calculus is utilized in real-world applications to project performance improvement.

Summation and Its Application

Summation is a key concept in calculus that deals with adding a sequence of numbers to find their aggregate. For this problem, summation is used to find the total production time for a series of units. The notation \( \sum \) is used to indicate the summation of a series of numbers which, in this exercise, corresponds to adding up times for each unit produced.

For example:

• To calculate the total time for producing units 1 through 10, we sum all \( T(x) \) from \( x=1 \) to \( x=10 \).
• The formula becomes \( \sum_{x=1}^{10} \left(2 + 0.3 \frac{1}{x}\right) \).

By systematically calculating each \( T(x) \) and summing them, we arrive at a total time of approximately 20.978690 hours. This represents practical use of summation in both academic and business settings, aiding in the evaluation of processes efficiency over time.

The Role of Unit Production

Unit production refers to the process of manufacturing individual units of a product. In this context, it's crucial due to its relation with time efficiency studies. The exercise showcases the concept of decreasing production time as more units are produced, which is typical in environments where workers become more adept with experience.

As we move on to calculating the time taken for producing units 20 through 30, we notice further refinement in time reductions. This is due to the experience curve and efficiency improvements becoming more significant as production continues.

Every unit from 20 to 30 takes slightly less time as calculated:

• For example, \( T(20) = 2.015 \) hours while \( T(30) = 2.01 \) hours.

The cumulative sum for these units indicates that the total time taken is about 22.137172 hours, showcasing how the time function effectively models such improvements. Understanding unit production in this way provides deeper insights into manufacturing processes and helps in planning for future production enhancements.