Problem 33 - The Calculus with Analytic Geometry

Question

For the production of a certain commodity, if \(x\) is the number of machines used and \(y\) is the number of man-hours, the number of units of the commodity produced is \(f(x, y)\) and \(f(x, y)=6 x y\). Such a function \(f\) is called a production function and the level curves of \(f\) are called constant product curves. Draw the constant product curves for this function \(f\) at \(30,24,18,12,6\), and 0 .

Step-by-Step Solution

Answer

The constant product curves are given by \(y = \frac{5}{x}\), \(y = \frac{4}{x}\), \(y = \frac{3}{x}\), \(y = \frac{2}{x}\), \(y = \frac{1}{x}\), and \(y = 0\).

Step 1: Identify the Constant Product Curves

Identify the constant levels at which the production function needs to be evaluated. In this case, the constant product levels are 30, 24, 18, 12, 6, and 0.

Step 2: Set the Production Function

Use the given production function: \[ f(x, y) = 6xy \]

Step 3: Formulate the Equations for Each Level Curve

For each constant product level, set \( f(x, y) = k \) where \( k \) is one of the given constants (30, 24, 18, 12, 6, 0). Solve for \( y \) in terms of \( x \): - For \( k = 30 \): \[ 6xy = 30 \implies y = \frac{30}{6x} = \frac{5}{x} \] - For \( k = 24 \): \[ 6xy = 24 \implies y = \frac{24}{6x} = \frac{4}{x} \] - For \( k = 18 \): \[ 6xy = 18 \implies y = \frac{18}{6x} = \frac{3}{x} \] - For \( k = 12 \): \[ 6xy = 12 \implies y = \frac{12}{6x} = \frac{2}{x} \] - For \( k = 6 \): \[ 6xy = 6 \implies y = \frac{6}{6x} = \frac{1}{x} \] - For \( k = 0 \): \[ 6xy = 0 \implies y = \frac{0}{6x} = 0 \]

Step 4: Draw the Level Curves

Plot each of the equations \( y = \frac{5}{x} \), \( y = \frac{4}{x} \), \( y = \frac{3}{x} \), \( y = \frac{2}{x} \), \( y = \frac{1}{x} \), and \( y = 0 \) on a graph. These curves show the relationship between the number of machines \( x \) and the number of man-hours \( y \) for the production of a constant number of units.

Key Concepts

Level Curves

Level curves, also known as contour lines, are essential in understanding functions of two variables. In this context, level curves represent combinations of machines and man-hours that produce the same quantity of goods. For the production function given by:
\[ f(x, y) = 6xy \]
the level curves can be obtained by setting the function equal to a constant value, such as 6, 12, 18, 24, and 30. Each level curve is derived by solving the equation \[ 6xy = k \]for different constants. For example, when \[ k = 30, \]solving \[ 6xy = 30 \]gives \[ y = \frac{30}{6x} = \frac{5}{x}.\]Similarly, other values for k yield different curves. These curves depict all the (x, y) pairs that result in the same production level.

Constant Product Curves

Constant product curves are another way to visualize the same concept of level curves. They show the pairs of variables that result in an unchanging output level. In the exercise, the production levels are constants like 30, 24, 18, 12, 6, and 0. Solving the production function for each level, we get equations like \[ y = \frac{5}{x} \]for \[ k = 30 \]and \[ y = \frac{4}{x} \]for \[ k = 24. \]Here, x and y represent the number of machines and man-hours respectively. By plotting these curves, you can see how various combinations of resources result in a fixed production level. The curves for higher constants are closer to the origin and as the constant decreases, the curves move further away.

Coordinate Graphing

To better understand the relationship between x and y, we often graph the constant product curves on a coordinate plane. This coordinate graphing helps us visualize the behavior of the production function. For example, plotting \[ y = \frac{5}{x}, y = \frac{4}{x}\]and so on, for different k values, shows us how man-hours change with the number of machines to maintain the same production level. On the graph, the x-axis represents the number of machines while the y-axis represents the number of man-hours. Each curve highlights different resource combinations to meet specific production levels. By comparing these curves, you can easily understand the efficiency and requirements for various production targets.

Solving Equations

Solving equations for constant product curves involves finding the values of y for given values of x and k. For instance, if we have the equation \[ 6xy = 18, \]we can solve for y by rearranging it: \[ y = \frac{18}{6x} = \frac{3}{x}. \]This process helps identify the pairs of machines and man-hours needed to maintain the same production output. Each step involves simple algebraic manipulation to isolate y. Doing this for every level curve specified, such as 6, 12, 18, 24, and 30, allows us to plot and analyze each curve individually. Ensure to understand each algebraic manipulation clearly, as this forms the basis for understanding the interplay between variables in the production function.