Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Manufacturers sometimes use empirically based formulas to predict the time required to produce the \(n\)th item on an assembly line for an integer \(n\). If \(T(n)\) denotes the time required to assemble the \(n\)th item and \(T_{1}\) denotes the time required for the first, or prototype, item, then typically \(T(n)=T_{1} n^{-k}\) for some positive constant \(k\). (a) For many airplanes, the time required to assemble the second airplane, \(T(2)\), is equal to \((0.80) T_{1}\). Find the value of \(k\). (b) Express, in terms of \(T_{1}\), the time required to assemble the fourth airplane. (c) Express, in terms of \(T(n)\), the time \(T(2 n)\) required to assemble the \((2 n)\) th airplane.

Step-by-Step Solution

Answer

(a) \(k \approx 0.322\); (b) \(T(4) \approx 0.698T_1\); (c) \(T(2n) = 0.80T(n)\).

Step 1: Understanding the Problem

We have a formula \(T(n) = T_1 n^{-k}\) where `T(n)` is the time to produce the nth item, \(T_1\) is the time for the first item, and \(k\) is a constant. We need to find \(k\) when \(T(2) = 0.80T_1\). Then, we need to calculate \(T(4)\) in terms of \(T_1\) and \(T(2n)\) in terms of \(T(n)\).

Step 2: Finding the Value of k

We start with the formula \(T(2) = T_1 \, 2^{-k}\). Given that \(T(2) = 0.80T_1\), we can set up the equation: \[0.80T_1 = T_1 \, 2^{-k}\]\[0.80 = 2^{-k}\]\[\ln(0.80) = -k\ln(2)\]. Solving gives \[-k = \frac{\ln(0.80)}{\ln(2)}\] and \(k \approx 0.322\).

Step 3: Calculating T(4)

Using the formula \(T(n) = T_1 n^{-k}\) with \(n = 4\) and \(k = 0.322\), we have: \[T(4) = T_1 \, 4^{-0.322}\]. Calculate \(4^{-0.322}\) to find \(T(4) \approx 0.698T_1\).

Step 4: Expressing T(2n) in Terms of T(n)

Using \(T(2n) = T_1 (2n)^{-k}\) and \(T(n) = T_1 n^{-k}\), we find: \[T(2n) = T_1 \, 2^{-k} n^{-k}\] which can be expressed as \(T(2n) = 2^{-k} T(n)\). With \(k \approx 0.322\), \(T(2n) = T(n) \, (0.80)\).

Key Concepts

Empirical Formulas

Empirical formulas are mathematical equations derived from real-world data rather than theoretical deductions. They are used extensively in various industries to estimate outcomes and streamline processes. In manufacturing, empirical formulas help predict how long it will take to produce an item after the first prototype is made. The formula provided, \( T(n) = T_1 n^{-k} \), is a common type of empirical formula. Here:

• \( T(n) \) is the time required for producing the \( n^{th} \) item.
• \( T_1 \) is the time needed for the first item, also known as the prototype item.
• \( n^{-k} \) indicates the learning rate or process efficiency improvement factor.

Such formulas assume that as more units are produced, the time taken decreases due to learning and efficiency improvements.

Assembly Time Calculation

Assembly time calculation involves using empirical formulas to determine the time required to produce each subsequent unit in a line of production. Given the formula \( T(n) = T_1 n^{-k} \), we can predict how much time will be required as more products are assembled.For example, when we were told that the time to assemble the second one \( T(2) = (0.80)T_1 \), it means the second unit is assembled 20% faster.To find the time for the fourth item, we plugged \( n = 4 \) into the formula resulting in \( T(4) = T_1 \, 4^{-0.322} \), obtaining an estimate of \( 0.698T_1 \). This shows gradual efficiency gains as production continues.

Production Efficiency

Production efficiency refers to the ability to produce maximum outputs with minimum input or waste. In the context of the assembly line model provided, it is reflected in the constant \( k \).In our formula, \( T(n) = T_1 n^{-k} \):

• Efficiency increases as \( k \) increases, reducing \( T(n) \) significantly for higher \( n \).
• When \( T(2) = 0.80T_1 \), the value of \( k \) was calculated as approximately 0.322. This indicates a 20% efficiency gain moving from the first to the second product.

Thus, higher production efficiency often results in quicker assembly times and reduced costs, due to better learning curves and process improvements.

Mathematical Modeling

Mathematical modeling is a method of simulating real-world processes to make predictions or decisions based on mathematical equations. In our exercise, the process of learning and improvement in an assembly line is represented by the formula \( T(n) = T_1 n^{-k} \).Some key points about mathematical modeling in this context:

• It allows estimation of production times based on prior data rather than suppositions alone.
• Provides a structured approach to improving manufacturing processes through quantifiable metrics.
• Models need to be regularly updated with new data to enhance predictions, ensuring they remain useful and accurate over time.

Models like this assist manufacturers in optimizing workflows and resources, ultimately driving efficiency and cost-effectiveness.