Problem 23 - Thomas Calculus

Question

Marginal cost Suppose that the dollar cost of producing \(x\) washing machines is \(c(x)=2000+100 x-0.1 x^{2} .\) a. Find the average cost per machine of producing the first 100 washing machines. b. Find the marginal cost when 100 washing machines are produced. c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Step-by-Step Solution

Answer

a. The average cost for 100 machines is $120. b. The marginal cost at 100 machines is $80. c. The cost of producing the 101st machine is approximately $79.90.

Step 1: Average Cost Formula

The average cost per machine is calculated by dividing the total cost function by the number of machines. The formula for average cost \( AC(x) \) is given by:\[AC(x) = \frac{c(x)}{x}\]Substitute \( c(x) = 2000 + 100x - 0.1x^2 \) to get:\[AC(x) = \frac{2000 + 100x - 0.1x^2}{x}\]

Step 2: Compute the Average Cost for 100 Machines

Substitute \( x = 100 \) into the average cost formula:\[AC(100) = \frac{2000 + 100(100) - 0.1(100)^2}{100}\]Calculate the numbers inside the brackets first:\[= \frac{2000 + 10000 - 1000}{100}\]Simplify:\[= \frac{12000}{100} = 120\]Therefore, the average cost per machine for producing 100 washing machines is 120 dollars.

Step 3: Marginal Cost Formula

Marginal cost \( MC(x) \) is the derivative of the total cost function \( c(x) \). Differentiate \( c(x) = 2000 + 100x - 0.1x^2 \):\[MC(x) = \, \frac{d}{dx}(2000 + 100x - 0.1x^2)\]This simplifies to:\[MC(x) = 100 - 0.2x\]

Step 4: Compute the Marginal Cost at 100 Machines

Substitute \( x = 100 \) into the marginal cost formula:\[MC(100) = 100 - 0.2(100)\]Calculate:\[= 100 - 20 = 80\]Thus, the marginal cost when producing 100 washing machines is 80 dollars.

Step 5: Cost of Producing the 101st Machine

Calculate the cost of producing 101 machines:\[c(101) = 2000 + 100(101) - 0.1(101)^2\]Evaluate:\[= 2000 + 10100 - 1020.1 = 12079.9\]Calculate the cost of producing 100 machines again (from Step 2):\[c(100) = 12000\]The cost of producing one more machine after 100 have been produced is:\[c(101) - c(100) = 12079.9 - 12000 = 79.9\]This result matches closely with the marginal cost of 80 dollars, confirming the approximation.

Key Concepts

Average Cost

Average cost is a helpful way to see the cost spread across each product you make. It tells us, on average, how much each unit costs to produce over a certain number of items.
A useful formula for finding the average cost is:

• \[ AC(x) = \frac{c(x)}{x} \]

where \( c(x) \) is the total cost function and \( x \) is the number of units.
In our exercise, the total cost to produce washing machines is given by the function \( c(x) = 2000 + 100x - 0.1x^2 \).
We use this formula to find the cost per washing machine when 100 units are produced:

• Plug 100 into the average cost formula: \[ AC(100) = \frac{2000 + 100(100) - 0.1(100)^2}{100} \]
• Calculating, we find that \( AC(100) = 120 \) dollars per washing machine.

This result helps businesses know the cost burden each additional item carries in bulk production.

Derivative

The derivative is like a magic wand in mathematics, which helps us find out how fast something is changing. It's used extensively to find the rate of change of a function with respect to its variables.
In economics, we frequently encounter derivatives when calculating marginal costs, which provide the cost change from producing one more unit.
For our specific cost function \( c(x) = 2000 + 100x - 0.1x^2 \), the derivative \( c'(x) \), or the marginal cost \( MC(x) \), is found as follows:

• The derivative of a constant like 2000 is 0.
• The derivative of \( 100x \) is 100.
• The derivative of \( -0.1x^2 \) is \( -0.2x \).

Combining these gives us:

• \[ MC(x) = 100 - 0.2x \]

This formula lets us calculate how the cost to produce changes as the number produced changes.

Cost Function

A cost function tells us everything about the production cost related to the number of items produced. Each part of the cost function provides different insights:

• The constant term, such as 2000 in \( c(x) = 2000 + 100x - 0.1x^2 \), often represents fixed costs which don't change with production volume.
• The linear term, \( 100x \), represents variable costs that change directly with the number of items produced.
• The quadratic term, \( -0.1x^2 \), can show additional complexities like discounts for bulk production or increased costs due to scale inefficiencies.

This function is the starting point for finding different types of cost-related information, such as average cost or marginal cost. Understanding it can help businesses plan their production and pricing strategies effectively.

Economics

Economics is all about making choices in a world of limited resources. It involves making decisions on the allocation of resources, balancing wants and needs, understanding how markets work, and predicting how changes in one part of the economy might impact another.
Within the context of cost and production, economics examines how companies can operate efficiently and remain competitive.

• It studies various cost structures, like fixed and variable costs, and how these affect pricing and supply decisions.
• By understanding concepts like average and marginal cost, businesses can make informed decisions on production levels.

For example, if producing additional units past a certain point leads to higher costs, a business must decide whether increasing production is profitable. Economics thus provides tools for analyzing production costs, aiding firms in decision-making to optimize revenue and minimize losses.