Find A Formula For Y In Terms Of X Given The Equation 100x^(3/4)y^(1/4) = 5400

Jun 21, 2025 by ADMIN 79 views

Optimizing Cell Phone Production Costs A Detailed Analysis

In the realm of manufacturing, the efficient allocation of resources is paramount to maximizing output and minimizing costs. This article delves into a scenario where a manufacturer aims to produce 5400 cell phones by strategically investing in labor and capital. The relationship between the expenditure on labor ( $x$ ) and capital ( $y$ ) is defined by the equation:

100x^(3/4)y^(1/4) = 5400

This mathematical model provides a framework for understanding how these two factors interact to achieve the desired production level. Our primary objective is to derive a formula that expresses $y$ in terms of $x$ , allowing us to analyze the capital investment required for various levels of labor expenditure. This exploration will not only provide valuable insights into the cost structure of cell phone production but also demonstrate the application of mathematical principles in real-world business scenarios.

Understanding the Production Equation

Before we embark on the algebraic manipulation, let's dissect the equation and understand its implications. The equation 100x^(3/4)y^(1/4) = 5400 is a Cobb-Douglas production function, a widely used economic model that describes the relationship between inputs (labor and capital) and output (cell phones). The exponents (3/4 and 1/4) represent the output elasticity of labor and capital, respectively. These elasticities indicate the percentage change in output for a 1% change in the corresponding input, holding other inputs constant.

In simpler terms, the exponent of 3/4 for labor ( $x$ ) suggests that a 1% increase in labor expenditure will lead to a 0.75% increase in cell phone production, assuming capital investment remains constant. Similarly, the exponent of 1/4 for capital ( $y$ ) implies that a 1% increase in capital expenditure will result in a 0.25% increase in production, with labor expenditure held constant. These exponents highlight the relative importance of labor and capital in the production process, with labor having a more significant impact on output in this particular scenario.

The constant 100 in the equation represents the total factor productivity (TFP), a measure of how efficiently inputs are converted into output. A higher TFP indicates better technology or management practices that enhance productivity. The value 5400 represents the target production level of cell phones.

Isolating $y$ in the Equation

To find a formula for $y$ in terms of $x$ , we need to isolate $y$ on one side of the equation. Let's break down the steps:

Divide both sides by 100:
```
x^(3/4)y^(1/4) = 54
```
This step simplifies the equation by removing the constant factor on the left side.
Isolate the term with $y$ :
```
y^(1/4) = 54 / x^(3/4)
```
We divide both sides by $x^(3/4)$ to isolate the term containing $y$ .
Raise both sides to the power of 4:
```
(y^(1/4))^4 = (54 / x^(3/4))^4
```
To eliminate the fractional exponent of 1/4 on $y$ , we raise both sides of the equation to the power of 4.
Simplify the equation:
```
y = 54^4 / (x^(3/4))^4
```
Applying the power of a power rule, we simplify the equation.
Further simplification:
```
y = 54^4 / x^3
```
We simplify the exponent in the denominator.
Calculate 54^4:
```
y = 8503056 / x^3
```
We calculate the value of 54 raised to the power of 4.

Therefore, the formula for $y$ in terms of $x$ is:

y = 8503056 / x^3

This equation reveals a crucial relationship: the amount of capital ( $y$ ) required is inversely proportional to the cube of the amount spent on labor ( $x$ ). This means that as labor expenditure increases, the required capital investment decreases dramatically. This inverse relationship highlights the potential for manufacturers to optimize their production costs by strategically balancing labor and capital investments.

Implications of the Formula

The formula y = 8503056 / x^3 provides a powerful tool for analyzing the trade-off between labor and capital in cell phone production. Let's delve into some of the key implications:

Cost Optimization: The formula allows manufacturers to determine the optimal combination of labor and capital expenditures to achieve the target production level of 5400 cell phones. By analyzing the cost of labor and capital, they can identify the point where the total cost is minimized.
Investment Decisions: The equation can guide investment decisions by quantifying the impact of changes in labor costs on capital requirements, and vice versa. For example, if labor costs increase, the manufacturer can use the formula to determine the necessary adjustments in capital investment to maintain the desired output.
Production Planning: The formula facilitates production planning by providing a clear relationship between labor, capital, and output. This allows manufacturers to forecast resource needs and adjust production schedules based on market demand and resource availability.
Technological Advancements: The formula can be used to assess the impact of technological advancements on the production process. For instance, the introduction of automation technologies might reduce the reliance on labor, leading to a shift in the optimal labor-capital mix.

Practical Applications and Examples

To illustrate the practical applications of the formula, let's consider a few examples:

Example 1: Suppose the manufacturer spends $100 on labor ($ x = 100$). Using the formula, we can calculate the required capital expenditure:

y = 8503056 / 100^3 = 8503056 / 1000000 = 8.503056

In this case, the manufacturer needs to spend approximately $8.50 on capital to produce 5400 cell phones.

Example 2: If the labor expenditure is increased to $200 ($ x = 200$), the required capital expenditure becomes:

y = 8503056 / 200^3 = 8503056 / 8000000 = 1.062882

As expected, the capital expenditure decreases significantly to approximately $1.06 when labor expenditure doubles. This demonstrates the inverse relationship between labor and capital.

Example 3: Let's say the manufacturer wants to reduce capital expenditure to $0.50 ($ y = 0.50$). We can rearrange the formula to solve for $x$ :

x^3 = 8503056 / y
x^3 = 8503056 / 0.50 = 17006112
x = (17006112)^(1/3) ≈ 257.13

Therefore, the manufacturer needs to spend approximately $257.13 on labor to reduce capital expenditure to $0.50.

These examples demonstrate how the formula can be used to analyze various scenarios and make informed decisions regarding resource allocation.

Conclusion

The formula y = 8503056 / x^3 provides a valuable tool for manufacturers seeking to optimize their production costs. By understanding the relationship between labor and capital expenditures, they can make informed decisions about resource allocation and achieve their production goals efficiently. This analysis underscores the importance of mathematical modeling in business decision-making, allowing for a more data-driven and strategic approach to resource management. The inverse relationship between labor and capital highlights the potential for cost savings through strategic investment, ultimately contributing to increased profitability and competitiveness in the dynamic cell phone market.

In conclusion, this detailed analysis provides a comprehensive understanding of how to optimize cell phone production costs by leveraging the relationship between labor and capital. The derived formula serves as a practical tool for manufacturers to make informed decisions, plan production effectively, and adapt to changing market conditions. By embracing a mathematical approach to resource management, manufacturers can enhance their efficiency, profitability, and overall success in the competitive landscape.