Problem 61 - College Algebra with Corequisite Support

Question

The cost in dollars of making \(x\) items is given by the function \(C(x)=10 x+500\) (a) The fixed cost is determined when zero items are produced. Find the fixed cost for this item. (b) What is the cost of making 25 items? (c) Suppose the maximum cost allowed is \(\$ 1500\). What are the domain and range of the cost function, \(C(x) ?\)

Step-by-Step Solution

Answer

(a) Fixed cost is \(\$500\). (b) Cost of 25 items is \(\$750\). (c) Domain: \([0, 100]\), Range: \([500, 1500]\).

Step 1: Calculate Fixed Cost

The fixed cost is determined when no items are produced, meaning when \(x = 0\). Substitute \(x = 0\) into the cost function \(C(x) = 10x + 500\). We have:\[C(0) = 10(0) + 500 = 500.\]Therefore, the fixed cost is \(\$500\).

Step 2: Calculate Cost of 25 Items

To find the cost of producing 25 items, substitute \(x = 25\) into the cost function \(C(x) = 10x + 500\). We have:\[C(25) = 10(25) + 500 = 250 + 500 = 750.\]Therefore, the cost of making 25 items is \(\$750\).

Step 3: Determine Domain and Range Based on Maximum Cost

Given that the maximum cost allowed is \(\$1500\), we need to calculate the maximum number of items that can be produced without exceeding this cost. Set the equation \(C(x) = 10x + 500\) equal to \(1500\):\[10x + 500 = 1500\]Subtract 500 from both sides:\[10x = 1000\]Divide by 10:\[x = 100\]The maximum number of items that can be produced is 100. Therefore, the domain of \(C(x)\) is \([0, 100]\).The range of \(C(x)\) is determined by the minimum and maximum costs. The minimum cost occurs when \(x = 0\) (fixed cost), which is \(500\). The maximum cost occurs when \(x = 100\), which is:\[C(100) = 10(100) + 500 = 1000 + 500 = 1500.\]Therefore, the range is \([500, 1500]\).

Key Concepts

Fixed Cost

In business and economics, a fixed cost refers to the expenses that do not vary with the level of goods or services produced. Regardless of the number of items made, you incur these costs. They could include things like rent, salaries, and machinery maintenance.
To determine the fixed cost from a cost function, set the production level, denoted by \( x \), to zero. This reflects the cost even when no output is generated. For the function \( C(x) = 10x + 500 \), substituting \( x = 0 \) reveals the fixed cost:

• \( C(0) = 10(0) + 500 \)
• \( = 0 + 500 \)
• \( = 500 \)

The fixed cost is therefore \( \$500 \). This cost remains constant regardless of the number of items produced.

Domain and Range

The domain and range of a function like a cost function help you understand the limits and possibilities of production costs. The domain refers to all the possible values of \( x \) (items produced) that make sense in the real world.
For the function \( C(x) = 10x + 500 \), the domain is limited by the maximum allowable cost, which is \( \\(1500 \). By calculating \( x \) when \( C(x) = 1500 \), we find:

• \( 10x + 500 = 1500 \)
• \( 10x = 1000 \)
• \( x = 100 \)

Thus, you can produce between 0 and 100 items. The domain is \([0, 100]\). The range focuses on the output, the cost, and starts from the fixed cost \( \\)500 \), reaching up to \( \$1500 \) when the production is maximized.
The range of \( C(x) \) can be found by looking at the minimum and maximum values, which leads to a range of \( [500, 1500] \).
These boundaries illustrate the expected cost spectrum given the production limits.

Function Evaluation

Evaluating a function means plugging a certain value into the function to find the corresponding output. In the context of cost functions, the specific value represents a number of items produced, while the function output is the cost of production.
For example, to find the cost of producing 25 items using \( C(x) = 10x + 500 \), substitute \( x = 25 \):

• \( C(25) = 10 \times 25 + 500 \)
• \( C(25) = 250 + 500 \)
• \( C(25) = 750 \)

This evaluation shows that the cost for manufacturing 25 items is \( \$750 \).
This process is fundamental not just in cost functions, but across all kinds of mathematical problems, serving as a tool to find specific results by inserting real-world numbers into functions.