Problem 87 - Brief Applied Calculus

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 50th Boeing 707 .

Step-by-Step Solution

Answer

The 50th Boeing 707 took 40,800 work-hours to build.

Step 1: Identify the Formula

The learning curve formula given is \(150n^{-0.322}\), where \(n\) is the number of units. It represents the work-hours in thousands needed to produce the \(n^{th}\) airplane.

Step 2: Insert the Unit Number

We need to find the work-hours for the 50th Boeing 707. Thus, replace \(n\) with 50 in the formula: \(150 \times 50^{-0.322}\).

Step 3: Calculate the Exponent

Compute \(50^{-0.322}\) using a calculator: \(50^{-0.322} \approx 0.272\).

Step 4: Multiply by the Coefficient

Now multiply the result by 150: \(150 \times 0.272\).

Step 5: Final Calculation

Perform the multiplication: \(150 \times 0.272 = 40.8\). Thus, the work-hours required is 40.8 thousand hours.

Key Concepts

Boeing 707

The Boeing 707 was a revolutionary aircraft introduced in the late 1950s. Marking a significant shift in aviation, it was one of the first successful commercial jetliners.
Before the 707, air travel was predominantly powered by propeller-driven aircraft. The introduction of the 707, with its jet engines, allowed for faster travel at higher altitudes, reducing both flight time and fuel consumption. This was a major leap forward for the aviation industry.
The Boeing 707's development and production highlighted the importance of learning curves in manufacturing. As production progressed, efficiency increased, leading to reduced work-hours for each successive airplane produced. This improvement was critical to reducing costs and making jet travel accessible to a broader audience. Understanding its production process helps illustrate key economic principles at play in mass manufacturing.

work-hours calculation

Calculating work-hours is essential when analyzing productivity and efficiency, especially in manufacturing. In the context of the Boeing 707 production, this is done using a learning curve formula which predicts the time needed for each succeeding unit.

• The formula: The given learning curve is expressed as \(150n^{-0.322}\), where \(n\) is the unit number.
• For the 50th Boeing 707: Insert \(n = 50\) into the formula.
• Calculate: \(150 \times 50^{-0.322}\).

The work-hours calculation involves computing this expression. It reflects how production becomes more efficient as more units are built, requiring fewer work-hours over time. Understanding this process helps in planning, resource allocation, and budgeting in manufacturing projects.

exponential functions

Exponential functions are mathematical expressions where variables appear as exponents. They describe various growth or decay processes.
In the learning curve formula \(150n^{-0.322}\), we see an exponential decay function. Here, the exponent \(-0.322\) signifies how much the work-hours decrease as more Boeing 707s are produced.

• Negative exponent: Signifies a decrease in the total work-hours as production increases.
• Base of the exponent: In this case, \(n\) represents the unit number, showing the position of the airplane in the production sequence.
• Result interpretation: As \(n\) increases, \(n^{-0.322}\) becomes smaller, leading to fewer required work-hours.

Understanding exponential functions is crucial for interpreting data in various fields, including economics, biology, and engineering. They provide insight into how quantities change rapidly or gradually under different conditions.