Problem 48 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

The output \(Q\) at a certain factory is related to inputs \(x\) and \(y\) by the equation $$ Q=x^{3}+2 x y^{2}+2 y^{3} $$ If the current levels of input are \(x=10\) and \(y=20\), use calculus to estimate the change in input \(y\) that should be made to offset an increase of \(0.5\) in input \(x\) so that output will be maintained at its current level.

Step-by-Step Solution

Answer

The change in input \(y\) should be approximately -0.171875.

Step 1: Recall the given function

The function for the output is given by \[ Q = x^3 + 2xy^2 + 2y^3 \]

Step 2: Calculate partial derivatives

Find the partial derivatives of the function with respect to both variables, x and y.\[ \frac{\partial Q}{\partial x} = 3x^2 + 2y^2 \]\[ \frac{\partial Q}{\partial y} = 4xy + 6y^2 \]

Step 3: Evaluate partial derivatives

Evaluate the partial derivatives at the given values \(x = 10\) and \(y = 20\).\[\frac{\partial Q}{\partial x} \bigg|_{x=10, y=20} = 3(10)^2 + 2(20)^2 = 3(100) + 2(400) = 300 + 800 = 1100\]\[ \frac{\partial Q}{\partial y} \bigg|_{x=10, y=20} = 4(10)(20) + 6(20)^2 = 800 + 2400 = 3200 \]

Step 4: Apply total differential approximation

Use the total differential \(dQ\) to maintain output at its current level.\[ dQ = \frac{\partial Q}{\partial x} dx + \frac{\partial Q}{\partial y} dy \]Given \(dQ = 0\), substitute \(dx = 0.5\) and the evaluated partial derivatives.\[ 0 = 1100 \cdot 0.5 + 3200 \cdot dy \]

Step 5: Solve for \(dy\)

Isolate \(dy\) in the equation.\[ 0 = 550 + 3200 \cdot dy \]\[ -550 = 3200 \cdot dy \]\[ dy = \frac{-550}{3200} = -\frac{11}{64} \] Thus,\[ dy = -0.171875 \]

Key Concepts

calculus

The problem above is a great example of how calculus can be applied in practical scenarios. Calculus allows us to find the rates at which things change. This concept is key, especially in problems that involve finding how changes in one variable affect another.
For instance, partial derivatives help us understand the rate of change of a function with respect to each variable independently. In the exercise, we first derived the partial derivatives of the function with respect to the input variables \(x\) and \(y\).
This is useful in many fields, from physics to economics, providing a mechanism to better understand dynamic systems.

total differential

The total differential is an important concept in multivariable calculus. It provides an approximation of how a small change in several variables affects a function. It is essentially the sum of the partial derivatives, each weighted by the small change in the corresponding variable.
In our exercise, we used the total differential to maintain the output \(Q\) at its current level despite changes in inputs. With a given increase in \(x\), we estimated the necessary change in \(y\) using the total differential formula:
\[ dQ = \frac{\partial Q}{\partial x} dx + \frac{\partial Q}{\partial y} dy \]
This allows us to maintain the balance in the system described by our function.

business and economics mathematics

In business and economics mathematics, understanding the relationships between different variables is essential. The exercise demonstrates how mathematical techniques can be applied to real-world scenarios, such as production processes.
Partial derivatives and the total differential are used to identify how changes in input factors can affect the output. For instance, in production, knowing how changes in labor input (like \(x\) and \(y\)) affect the total output (\(Q\)) can aid in decision-making.
This allows businesses to efficiently allocate resources, maximize output, and sustain productivity levels in response to varying inputs. Mathematics provides a structured way to analyze and optimize these important decisions.

input-output analysis

Input-output analysis is a method used to understand the interactions between different components of a system, such as inputs and outputs in an economic context.
In the provided exercise, we looked at how adjusting the input variables (\(x\) and \(y\)) will affect the output (\(Q\)). This kind of analysis helps businesses and economists understand the dependencies and sensitivities within a production process.
By analyzing the input-output relationship, better forecasting and planning can be achieved. Businesses can pinpoint which inputs can be adjusted to meet production targets or how to counteract fluctuations within the system. These insights are fundamental in making informed operational and strategic decisions.