Problem 96 - Applied Calculus

Question

BUSINESS: Learning Curves in Airplane Production Recall (pages \(26-27\) ) that the learning curve for the production of Boeing 707 airplanes is \(150 n^{-0.322}\) (thousand work-hours). Find how many work-hours it took to build: $$ \text { The } 250 \text { th Boeing } 707 \text { . } $$

Step-by-Step Solution

Answer

About 39.4 thousand work-hours are required for the 250th airplane.

Step 1: Identify the Learning Curve Formula

The learning curve formula for Boeing 707 airplanes is given by \( f(n) = 150n^{-0.322} \), where \( f(n) \) represents the work-hours required to build the \( n \)-th airplane.

Step 2: Substitute the Specific Airplane Number

We need to find the work-hours for the \( 250 \)-th airplane. Substitute \( n = 250 \) into the formula: \( f(250) = 150 \times 250^{-0.322} \).

Step 3: Calculate the Exponent

First, calculate \( 250^{-0.322} \). This requires finding the power of 250 raised to \(-0.322\).

Step 4: Compute the Work-Hours

Multiply the result from Step 3 by 150 to find the work-hours: \( f(250) = 150 imes (250^{-0.322}) \). Carry out the computation to get the final number of work-hours.

Step 5: Interpretation and Conclusion

The calculated value is the number of work-hours taken to produce the 250th Boeing 707. This reflects the efficiency gained through the learning curve.

Key Concepts

Understanding the Learning Curve

In a production setting, the learning curve is a concept used to describe how labor efficiency improves with experience. As more units are produced, workers often become more skilled, processes improve, and the time required to manufacture each unit decreases. This increase in efficiency can be mathematically modeled using a learning curve equation. In this case, the function for Boeing 707 production is expressed as \(150n^{-0.322}\). Here, \(n\) represents the sequential number of airplanes produced, while the exponent \(-0.322\) indicates the rate of learning and productivity improvement. When analyzing the function, one can understand that as \(n\) increases, the power is negative, leading to a smaller number, reflecting reduced work-hours per unit. This principle is significant in manufacturing as it serves as a predictive model for production planning. It helps industries project future labor needs and optimize their processes.

The History and Impact of Boeing 707 Production

The Boeing 707 was a landmark in aviation history, as it represented a leap forward in jetliner technology in the late 1950s. It was not only a breakthrough in terms of speed and range but also marked a substantial shift in manufacturing practices due to high demand. Producing such a complex aircraft involved innovations in design and manufacturing approaches, which were quantified using learning curve models.
For Boeing, the learning curve was crucial in ensuring the cost-efficient scaling of 707 production. As the production line matured, Boeing could decrease the work-hours needed for each subsequent aircraft, thus lowering costs and improving delivery times.

• This resulted in more competitive pricing.
• It also improved profit margins on each sale.
• Ultimately, it contributed to Boeing's leadership in the aviation market.

The application of the learning curve in Boeing's production strategy highlights its effectiveness as a management tool that drives continuous improvement and operational excellence.

Calculating Work-Hours in Production

To calculate the work-hours required for a specific unit in a production series, it's essential to apply the learning curve equation appropriately. The process involves using the given formula and substituting the unit number within it. For the Boeing 707, the equation is \( f(n) = 150n^{-0.322} \), specifically used to calculate work-hours for the \(n\)-th airplane. Let's break down the calculation for the 250th Boeing 707.

• First, substitute \( n = 250 \) into the formula to get \( f(250) = 150 \times 250^{-0.322} \).
• Next, compute \( 250^{-0.322} \). This involves finding the result of raising 250 to the power of \(-0.322\).
• Finally, multiply the resulting value by 150 to find the total work-hours required. This value shows how many thousands of work-hours are needed for the 250th unit.

This step-by-step calculation process underscores how the learning curve allows for accurate forecasting of labor needs as production progresses. It's crucial for ensuring proper resource allocation and for keeping production schedules on track.