Problem 9 - The Calculus with Analytic Geometry

Question

The fixed overhead expense of a manufacturer of children's toys is \(\$ 400\) per week, and other costs amount to \(\$ 3\) for each toy produced. Find (a) the total cost function, (b) the average cost function, and (c) the marginal cost function. (d) Show that there is no absolute minimum average unit cost. (e) What is the smallest number of toys that must be produced so that the average cost per toy is less than \(\$ 3.42 ?\) (f) Draw sketches of the graphs of the functions in (a), (b), and (c) on the same set of axes.

Step-by-Step Solution

Answer

a) \( C(x) = 400 + 3x \) b) \( AC(x) = \frac{400}{x} + 3 \) c) \( MC(x) = 3 \) d) No absolute minimum e) 953 toys f) Sketch graphs.

Step 1: Define the variables

Let the variable \( x \) represent the number of toys produced.

Step 2: Determine the fixed costs

The fixed overhead expense is given as \( \$400 \) per week.

Step 3: Determine the variable costs

The variable cost per toy is given as \( \$3 \). Therefore, the total variable cost for producing \( x \) toys is \( 3x \) dollars.

Step 4: Find the total cost function

The total cost function \( C(x) \) is the sum of the fixed costs and the variable costs: \[ C(x) = 400 + 3x \]

Step 5: Find the average cost function

The average cost function \( AC(x) \) is the total cost divided by the number of toys produced: \[ AC(x) = \frac{C(x)}{x} = \frac{400 + 3x}{x} = \frac{400}{x} + 3 \]

Step 6: Find the marginal cost function

The marginal cost function \( MC(x) \) is the derivative of the total cost function with respect to \( x \): \[ MC(x) = C'(x) = 3 \]

Step 7: Show no absolute minimum average unit cost

The average cost function \( AC(x) = \frac{400}{x} + 3 \) does not have an absolute minimum because as \( x \) approaches infinity, \( \frac{400}{x} \to 0 \), but it never reaches zero. Thus, the average cost function decreases indefinitely.

Step 8: Find the smallest number of toys for average cost < $3.42

Set the average cost function less than \$3.42: \[ \frac{400}{x} + 3 < 3.42 \] Subtract 3 from both sides: \[ \frac{400}{x} < 0.42 \] Multiply both sides by \( x \): \[ 400 < 0.42x \] Divide both sides by 0.42: \[ x > \frac{400}{0.42} \approx 952.38 \] Since the number of toys must be a whole number, \[ x \geq 953 \]

Step 9: Sketch the graphs

Draw the graphs of \( C(x) = 400 + 3x \) (a straight line starting at 400 and with slope 3), \( AC(x) = \frac{400}{x} + 3 \) (a hyperbola asymptoting to 3), and \( MC(x) = 3 \) (a horizontal line at y=3) on the same set of axes.

Key Concepts

total cost function

Understanding the total cost function is crucial in economics and business.
It represents the combined cost of producing a certain number of goods.
For this exercise, the total cost function is given by:
\[ C(x) = 400 + 3x \]

• **Fixed Costs:** These are costs that do not change with the number of toys produced.
  In this case, the fixed cost is \( \$400 \) per week.
• **Variable Costs:** These depend on the number of toys produced.
  Here, the variable cost is \( \$3 \) per toy, so the total variable cost for \( x \) toys is \( 3x \) dollars.

Adding these together gives the total cost.
Understanding this helps in planning and budgeting for production.

average cost function

The average cost function helps businesses understand the cost per unit.
It is calculated by dividing the total cost by the number of units produced.
For this example, the average cost function is:
\[ AC(x) = \frac{400 + 3x}{x} = \frac{400}{x} + 3 \]

This function shows how the average cost per toy changes with the number of toys produced.

• **High Initial Costs:** When only a few toys are produced, the average cost is high because the fixed costs are spread over fewer units.
• **Decreasing Average Costs:** As production increases, the fixed costs are spread over more toys, making the average cost decrease.

The average cost never has an absolute minimum because it continues to decrease as more toys are produced.

marginal cost function

The marginal cost function is critical for decision-making in production.
It shows the cost of producing one additional unit.
In this exercise, the marginal cost function is simply:
\[ MC(x) = 3 \]
This is the derivative of the total cost function.

• **Constant Marginal Cost:** Here, the marginal cost is constant at \( \$3 \) per toy.
  This means each additional toy costs the same to produce, regardless of the number of toys already produced.

Understanding the marginal cost helps in pricing and optimizing production levels.

calculus optimization

Calculus techniques are powerful tools for optimization in cost analysis.
In this exercise, we use calculus to analyze costs and determine the best production quantities.

• **Derivative:** The derivative of the total cost function gives us the marginal cost.
  It shows how much the total cost changes with the production of one more toy.
• **Extreme Values:** In some cases, we look for minimum or maximum values to optimize costs.
  However, the average cost function \( AC(x) = \frac{400}{x} + 3 \) decreases indefinitely and has no absolute minimum.

This helps businesses minimize costs and maximize profits effectively.

cost analysis

Cost analysis involves evaluating the different costs associated with production.
This exercise provides a comprehensive look at various cost functions.

• **Total Cost Analysis:** Combining fixed and variable costs to understand total production costs.
• **Average Cost Analysis:** Evaluating the cost per unit to aid in pricing and cost management.
• **Marginal Cost Analysis:** Understanding the cost of producing one more unit, crucial for making production decisions.
• **Optimization:** Using calculus to find optimal production levels to minimize costs.
  For instance, to find the number of toys needed for the average cost to be less than \( \$3.42 \), we set \( \frac{400}{x} + 3 < 3.42 \).

By conducting a thorough cost analysis, businesses can make informed decisions that enhance efficiency and profitability.