Problem 88
Question
Recall (pages \(27-28)\) that the learning curve for the production of Boeing 707 airplanes is \(150 \mathrm{n}^{-0.322}\) (thousand work-hours). Find how many work-hours it took to build: The 250th Boeing 707 .
Step-by-Step Solution
Verified Answer
The 250th Boeing 707 took approximately 39.78 thousand work-hours to build.
1Step 1: Understand the Formula
The given learning curve formula is used to determine the number of work-hours required to produce an item based on its production number. The formula is \(150 \cdot n^{-0.322}\), where \(n\) is the production order of the airplane.
2Step 2: Substitute the Production Number
We need to find the work-hours for the 250th Boeing 707. Substitute \(n = 250\) into the formula: \(150 \cdot (250)^{-0.322}\).
3Step 3: Calculate the Exponent
Calculate the exponent part of the formula: \(250^{-0.322} \approx 0.2652\).
4Step 4: Multiply by the Coefficient
Now, multiply the result of the exponent calculation by 150: \(150 \times 0.2652 = 39.78\).
5Step 5: Round if Necessary
Interpret the result as the approximate number of work-hours, which can be rounded to the nearest applicable measure based on context. In this case, keep it as \(39.78\) thousand work-hours as exact numbers may not be rounded in production settings.
Key Concepts
Work-Hours CalculationExponentiationProduction OrderEngineering Mathematics
Work-Hours Calculation
In engineering and production, determining the time needed for tasks is essential. Work-hours calculation helps estimate how much time an action will take. It's a core part of managing resources and schedules.
When dealing with repetitive tasks or producing large quantities of a product, the learning curve effect comes into play. Over time, workers become more efficient, and the time needed to complete a task decreases. Calculating work-hours helps optimize this process.
To use the learning curve formula, note the number of items already produced. In our example, the formula is structured as \(150\, n^{-0.322}\) (thousand work-hours), where \(n\) is the production order of the item. By substituting the specific production number into the formula, we can find the work-hours required for that particular item.
When dealing with repetitive tasks or producing large quantities of a product, the learning curve effect comes into play. Over time, workers become more efficient, and the time needed to complete a task decreases. Calculating work-hours helps optimize this process.
To use the learning curve formula, note the number of items already produced. In our example, the formula is structured as \(150\, n^{-0.322}\) (thousand work-hours), where \(n\) is the production order of the item. By substituting the specific production number into the formula, we can find the work-hours required for that particular item.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: a base and an exponent. It is written as \(a^b\), where \(a\) is the base and \(b\) is the exponent. This operation is a form of repeated multiplication.
In the context of learning curves, exponentiation is crucial because it dictates how the time decreases as production processes advance. A negative exponent signifies a reduction effect in the context of learning curves, meaning each successive unit takes less time to produce.
For example, in the formula \(150\, n^{-0.322}\), the \(n^{-0.322}\) part indicates that as \(n\), or the production number, increases, the time per unit reduces. This exponential decrease is the essence of the learning curve, showcasing efficient progression in production.
In the context of learning curves, exponentiation is crucial because it dictates how the time decreases as production processes advance. A negative exponent signifies a reduction effect in the context of learning curves, meaning each successive unit takes less time to produce.
For example, in the formula \(150\, n^{-0.322}\), the \(n^{-0.322}\) part indicates that as \(n\), or the production number, increases, the time per unit reduces. This exponential decrease is the essence of the learning curve, showcasing efficient progression in production.
Production Order
Production order plays a crucial role in understanding the dynamics of manufacturing. It refers to the sequence or number of items produced in a production line. Each item's position in this sequence can influence the efficiency and required work-hours due to factors like learning curves.
Higher production orders, as seen in \(n\) values in learning curve equations, typically represent reduced work-hours per item. This is because as workers continue to perform similar tasks, they become faster and more proficient. In our case, calculating for the 250th Boeing 707 provides insight into how production efficiency is achieved over time.
Understanding production order helps in planning and improving manufacturing processes. It allows noticing patterns, predicting future performance, and setting realistic timelines based on past data.
Higher production orders, as seen in \(n\) values in learning curve equations, typically represent reduced work-hours per item. This is because as workers continue to perform similar tasks, they become faster and more proficient. In our case, calculating for the 250th Boeing 707 provides insight into how production efficiency is achieved over time.
Understanding production order helps in planning and improving manufacturing processes. It allows noticing patterns, predicting future performance, and setting realistic timelines based on past data.
Engineering Mathematics
Engineering mathematics is the application of mathematical principles to solve engineering problems. It involves various advanced topics like calculus, differential equations, linear algebra, and more.
For production and efficiency analyses, such as in the learning curve for Boeing 707 airplanes, mathematical models are crucial for predicting outcomes and optimizing processes. The learning curve is just one example of how mathematical concepts apply to real-world engineering scenarios.
In this context, understanding functions, exponentiation, and interpreting mathematical results are key components. Engineering mathematics not only aids in solving current problems but also in forecasting future complexities, helping engineers innovate and improve systems continuously.
For production and efficiency analyses, such as in the learning curve for Boeing 707 airplanes, mathematical models are crucial for predicting outcomes and optimizing processes. The learning curve is just one example of how mathematical concepts apply to real-world engineering scenarios.
In this context, understanding functions, exponentiation, and interpreting mathematical results are key components. Engineering mathematics not only aids in solving current problems but also in forecasting future complexities, helping engineers innovate and improve systems continuously.
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