Problem 62
Question
BUSINESS: Marginal Average Cost A company can produce computer flash memory devices at a cost of \(\$ 6\) each, while fixed costs are \(\$ 50\) per day. Therefore, the company's cost function is \(C(x)=6 x+50\) a. Find the average cost function \(A C(x)=\frac{C(x)}{x}\) b. Find the marginal average cost function \(M A C(x)\). c. Evaluate \(M A C(x)\) at \(x=25\) and interpret your answer.
Step-by-Step Solution
Verified Answer
The marginal average cost at 25 units is -$0.08, meaning the average cost decreases by $0.08 per additional unit produced.
1Step 1: Write the Cost Function
The cost function is given as \(C(x) = 6x + 50\), where \(x\) represents the number of units produced per day.
2Step 2: Calculate the Average Cost Function
The average cost function \(AC(x)\) is calculated as \(\frac{C(x)}{x}\). Substitute the cost function into this formula: \[AC(x) = \frac{6x + 50}{x} = 6 + \frac{50}{x}\].
3Step 3: Differentiate to Find Marginal Average Cost
The marginal average cost function \(MAC(x)\) is the derivative of the average cost function \(AC(x)\). Differentiate \(AC(x) = 6 + \frac{50}{x}\) with respect to \(x\):\[MAC(x) = \frac{d}{dx}(6 + \frac{50}{x}) = 0 - \frac{50}{x^2} = -\frac{50}{x^2}\].
4Step 4: Evaluate at \(x=25\)
Substitute \(x = 25\) into the marginal average cost function: \[MAC(25) = -\frac{50}{25^2} = -\frac{50}{625} = -\frac{1}{12.5} = -0.08\]. The marginal average cost at 25 units is approximately -0.08.
5Step 5: Interpret the Result
The result \(MAC(25) = -0.08\) means that if production increases from 25 units, the average cost per unit decreases by approximately \$0.08.
Key Concepts
Cost FunctionAverage Cost FunctionDifferentiation
Cost Function
The cost function is a mathematical relationship that describes how total production costs change with varying levels of output. For a company producing flash memory devices, the cost function is defined as \( C(x) = 6x + 50 \). Here, \( x \) is the number of units produced, \( 6x \) represents the variable cost per unit, and 50 is the fixed cost, which remains constant regardless of production volume.
The variable cost depends on the number of units produced because it reflects the per-unit production expense. In this case, each flash memory device costs \\(6 to produce.
A deeper understanding of the cost function is crucial because it shows how costs behave as production scales up or down and helps businesses plan their finances more strategically.
The variable cost depends on the number of units produced because it reflects the per-unit production expense. In this case, each flash memory device costs \\(6 to produce.
A deeper understanding of the cost function is crucial because it shows how costs behave as production scales up or down and helps businesses plan their finances more strategically.
- Fixed Costs: These are costs that do not change with the level of output, for example, rent, salaries, or equipment costs. Here it is \\)50.
- Variable Costs: These costs vary with the production volume. Here it is \$6 per unit.
Average Cost Function
The average cost function helps businesses understand the cost per unit of output. It is derived by dividing the total cost \( C(x) \) by the number of units \( x \). For our scenario, this function is given by:
\[ AC(x) = \frac{C(x)}{x} = \frac{6x + 50}{x} = 6 + \frac{50}{x} \]
This equation informs us that, aside from the constant variable cost of \$6 per unit, the fixed cost's contribution to the average cost decreases as more units are produced. The term \( \frac{50}{x} \) signifies that as \( x \) increases, \( \frac{50}{x} \) approaches zero. Thus, spreading the fixed cost over more units reduces the average cost per unit.
Understanding the average cost structure aids companies in pricing strategies and identifying the output level at which they can achieve economies of scale.
\[ AC(x) = \frac{C(x)}{x} = \frac{6x + 50}{x} = 6 + \frac{50}{x} \]
This equation informs us that, aside from the constant variable cost of \$6 per unit, the fixed cost's contribution to the average cost decreases as more units are produced. The term \( \frac{50}{x} \) signifies that as \( x \) increases, \( \frac{50}{x} \) approaches zero. Thus, spreading the fixed cost over more units reduces the average cost per unit.
Understanding the average cost structure aids companies in pricing strategies and identifying the output level at which they can achieve economies of scale.
Differentiation
Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point. In the context of economics, it helps in determining how cost, revenue, or other functions are changing as output levels vary.
To find the marginal average cost function, we differentiate the average cost function \( AC(x) = 6 + \frac{50}{x} \) with respect to \( x \). The result of this differentiation gives:
\[ MAC(x) = \frac{d}{dx}(6 + \frac{50}{x}) = 0 - \frac{50}{x^2} = -\frac{50}{x^2} \]
The marginal average cost \( MAC(x) \) shows the rate of change of the average cost as production changes. A negative \( MAC(x) \) indicates that increasing production results in a decrease in average costs, highlighting economies of scale. At \( x=25 \), \( MAC(25) = -0.08 \) which illustrates that producing one additional unit at this level will decrease average costs by about \$0.08.
Through differentiation, businesses gain insights into optimal production strategies by understanding how costs adjust with changes in output.
To find the marginal average cost function, we differentiate the average cost function \( AC(x) = 6 + \frac{50}{x} \) with respect to \( x \). The result of this differentiation gives:
\[ MAC(x) = \frac{d}{dx}(6 + \frac{50}{x}) = 0 - \frac{50}{x^2} = -\frac{50}{x^2} \]
The marginal average cost \( MAC(x) \) shows the rate of change of the average cost as production changes. A negative \( MAC(x) \) indicates that increasing production results in a decrease in average costs, highlighting economies of scale. At \( x=25 \), \( MAC(25) = -0.08 \) which illustrates that producing one additional unit at this level will decrease average costs by about \$0.08.
Through differentiation, businesses gain insights into optimal production strategies by understanding how costs adjust with changes in output.
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