Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Average Cost The cost \(C\) (in dollars) of producing \(\underline{x}\) basketballs is \(C=375,000+4 x\). The average cost \(\bar{C}\) per basketball is \(\bar{C}=\frac{C}{x}=\frac{375,000+4 x}{x}, x>0\) (a) Sketch the graph of the average cost function. (b) Find the average costs of producing \(1000,10,000\), and 100,000 basketballs. (c) Find the horizontal asymptote and explain its meaning in the context of the problem.

Step-by-Step Solution

Answer

The plot of the average cost function shows a downward trend, indicating that the average cost decreases as the number of basketballs produced increases. The average costs for producing 1,000, 10,000, and 100,000 basketballs can be calculated using the given formula. The horizontal asymptote of the function is $4, indicating that the average cost per basketball will not go below $4, regardless of how many basketballs are produced.

Step 1: Graphing the Average Cost Function

In order to sketch the graph you can utilise plotting tools. On the x-axis, you can place the number of units produced \(x\), and on the y-axis, the average cost \(\bar{C}\). Based on the given formula \(\bar{C}=\frac{375,000+4 x}{x}\), plot the points on the graph where \(x>0\).

Step 2: Calculating Average Cost

For part (b), you'll need to find the average costs of producing 1,000, 10,000, and 100,000 basketballs. Use the given formula \(\bar{C}=\frac{375,000+4 x}{x}\) and substitute \(x\) with 1,000; 10,000; 100,000 respectively to determine the average cost for each number of units produced.

Step 3: Finding the Horizontal Asymptote

The horizontal asymptote of a function can be obtained by looking at the 'end behavior' of the function. As \(x\) approaches infinity, the fraction \(375,000/x\) approaches zero, leaving \(4\) as the horizontal asymptote. This implies that the average cost per basketball will approach $4 as the number of basketballs produced increases indefinitely. This suggests, in terms of the context of the problem, that no matter how many basketballs are produced, the cost per basketball (average cost) will not go below $4.

Key Concepts

Horizontal Asymptote

When we deal with functions, especially in the context of cost calculations, understanding horizontal asymptotes can be crucial. A horizontal asymptote represents a value that the function approaches as the input grows larger. In simpler terms, it's like a ceiling or floor that the function gets closer to but never crosses. With the average cost function \[ \bar{C} = \frac{375,000 + 4x}{x} \],as we increase the number of basketballs produced, say towards infinity,

• The term \( \frac{375,000}{x} \) diminishes.
• We're left with \( 4 \), the horizontal asymptote.

This asymptote suggests that the average production cost won't fall below \(4 per basketball, regardless of how many are made. This is because larger production spreads the fixed cost (\)375,000) over more units, minimizing its impact on unit cost. Horizontal asymptotes hence give us insight into the long-term behavior of cost functions.

Graphing Functions

Graphing the average cost function helps to visually understand cost behavior across production scales. To do this:

• Place the number of basketballs, \(x\), on the x-axis.
• Plot the average cost, \( \bar{C} \), on the y-axis.

Using the function \[ \bar{C} = \frac{375,000 + 4x}{x} \], plot crucial points such as where \(x = 1000\), \(x = 10,000\), and \(x = 100,000\). This provides a line graph showing how costs average out as production increases.
As the graph is plotted, observe:

• With low production, average costs are high due to the fixed cost's greater impact.
• As \(x\) increases, the average cost curve slopes downward, approaching the horizontal asymptote at \(4\).

Graphing tools or software can make this visualization easier, allowing adjustments to see potential cost scenarios.

Production Cost Calculation

Producing basketballs involves calculating both fixed and variable costs. Fixed costs, like the initial setup at \(375,000, remain constant regardless of production volume. Variable costs, on the other hand, depend on the number of units produced. Here, it's \)4 per basketball.

• Total cost formula: \( C = 375,000 + 4x \)
• Average cost formula: \( \bar{C} = \frac{375,000 + 4x}{x} \)

To understand this, compute the average cost for set production levels (e.g., \(x=1000\), \(x=10,000\), \(x=100,000\)). These calculations reveal how costs behave:

• At low \(x\), average costs are high due to fixed cost influence.
• With increasing \(x\), fixed costs significantly dilute, lowering the average.

Thus, understanding production cost calculations aids in strategizing manufacturing scale to optimize expenses and profitability.