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Chapter 2: Problem 18

The background for this exercise can be found in Exercises 11, 12, 13, and 14 in Section 1.4. A manufacturer of widgets has fixed costs of \(\$ 1200\) per month, and the variable cost is \(\$ 40\) per widget (so it costs \(\$ 40\) to produce 1 widget). Let \(N\) be the number of widgets produced in a month. a. Find a formula for the manufacturer's total cost \(C\) as a function of \(N\). b. The highest price \(p\), in dollars, of a widget at which \(N\) widgets can be sold is given by the formula \(p=53-0.01 N\). Using this, find a formula for the total revenue \(R\) as a function of \(N\). c. Use your answers to parts a and \(b\) to find \(a\) formula for the profit \(P\) of this manufacturer as a function of \(N\). d. Use your formula from part c to determine the two break-even points for this manufacturer. Assume here that the manufacturer produces the widgets in blocks of 50 , so a table setup showing \(N\) in multiples of 50 is appropriate. e. Use your formula from part c to determine the production level at which profit is maximized if the manufacturer can produce at most 1500 widgets in a month. As in part d, assume that the manufacturer produces the widgets in blocks of 50 .

Short Answer

Expert verified
a. \(C = 1200 + 40N\). b. \(R = 53N - 0.01N^2\). c. \(P = -0.01N^2 + 13N - 1200\). d. Break-even at \(N = 50\). e. Profit maximized at \(N = 650\).

Step by step solution

01

Develop Total Cost Formula

The manufacturer's total cost \(C\) is composed of fixed costs and variable costs. The fixed costs are \\(1200 per month. The variable cost is \\)40 per widget, and with \(N\) being the number of widgets, the variable cost becomes \(40N\). Thus, the total cost is given by the formula:\[C = 1200 + 40N\]
02

Formulate Total Revenue

The highest price \(p\) to sell a widget is \(p = 53 - 0.01N\). The total revenue \(R\) from selling \(N\) widgets is the price \(p\) times the quantity \(N\). Thus, the revenue function is:\[R = N(53 - 0.01N) = 53N - 0.01N^2\]
03

Determine Profit Formula

Profit \(P\) is the total revenue minus total cost. From Step 1 and Step 2, substitute \(R = 53N - 0.01N^2\) and \(C = 1200 + 40N\) into the profit equation \(P = R - C\):\[P = (53N - 0.01N^2) - (1200 + 40N)\]Simplifying gives:\[P = -0.01N^2 + 13N - 1200\]
04

Find Break-Even Points

Break-even occurs when profit \(P = 0\). Set the profit equation from Step 3 to zero and solve for \(N\):\[-0.01N^2 + 13N - 1200 = 0\]Using the quadratic formula \(N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -0.01\), \(b = 13\), and \(c = -1200\), calculate:\[N = \frac{-13 \pm \sqrt{13^2 - 4(-0.01)(-1200)}}{2(-0.01)} = 60 \text{ or } 2000\]Considering widgets are produced in blocks of 50, compile values of \(N\) in a table of multiples of 50. The feasible break-even points within production limits are \(50\) and \(2000\). However, \(N = 2000\) is beyond production capability.
05

Profit Maximization Check

To maximize profit, identify where the vertex of the quadratic function in \(P = -0.01N^2 + 13N - 1200\) is. The vertex form is at \(N = -\frac{b}{2a}\):\[N = \frac{-13}{2(-0.01)} = 650\]Check values in increments of 50 (multiples of production blocks) near 650 to confirm it maximizes profit within production limits:- For \(N = 650\), calculate \(P = -0.01(650)^2 + 13(650) - 1200\).- Confirm by plugging in surrounding multiples like \(600\) and \(700\) for comparison.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fixed Costs
Fixed costs are expenses that do not change with the level of production or sales volume. They are constant and include expenses such as rent, salaries, and insurance. For the widget manufacturer, fixed costs amount to \( \$1200 \) per month.

- These costs remain the same whether the manufacturer produces 0 widgets or 1,000 widgets.
- Even with no production, the company incurs these fixed costs. - Fixed costs ensure the company covers its basic overhead to keep the business running.

It's essential to distinguish these from variable costs, which fluctuate with production volume.
Variable Costs
Variable costs change with the number of units produced. For the widget manufacturer, each widget costs \( \\(40 \) to produce. Therefore, the variable costs increase as production increases. This cost is directly proportional to the number of widgets, represented as \( 40N \) in the total cost formula.

- If the company produces 100 widgets, the variable cost is \( 40 \times 100 = \\)4000 \).
- If no widgets are produced, the variable cost is \( \$0 \).
- Variable costs include materials, direct labor, and any additional expenses directly related to manufacturing.

Understanding both fixed and variable costs is crucial for managers to make informed production and pricing decisions.
Revenue Function
The revenue function relates to how much money the company earns from selling its products. To find the total revenue, multiply the price per widget by the number of widgets sold, \( N \). The price function in this example is \( p = 53 - 0.01N \), where \( p \) is the price per widget.

The revenue function, therefore, becomes:
\[ R = N(53 - 0.01N) = 53N - 0.01N^2 \]

- As \( N \) increases, each widget's price decreases because of the demand effect. - The function reflects diminishing returns on each additional widget sold.

This quadratic relationship highlights the need to balance the number of units sold with the decreasing price to maximize revenue.
Profit Maximization
Profit maximization is about finding the production level where profits are at their highest. Profit is calculated by subtracting total costs from total revenue. In this case, the profit function is \( P = -0.01N^2 + 13N - 1200 \).
  • The negative coefficient of \( N^2 \) indicates a parabolic shape with a maximum point.
  • To find the maximum profit, determine the vertex of the parabola using \( N = -\frac{b}{2a} \), which gives \( N = 650 \).
  • Producing 650 widgets maximizes profit, assuming capacity allows it.
Verifying by calculating profits for widget blocks near 650 ensures accuracy in finding the optimal production volume.

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