Problem 61 - College Algebra

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: $500 b. Cost for 25 items: $750 c. Domain: \(x \geq 0\), Range: \(500 \leq C(x) \leq 1500\)

Step 1: Understand the Fixed Cost

The fixed cost is the part of the total cost that does not depend on the number of items produced, which is essentially the cost when producing zero items. To find this, substitute \(x = 0\) in the cost function \(C(x) = 10x + 500\).

Step 2: Calculate the Fixed Cost

Substitute \(x = 0\) into the equation: \(C(0) = 10(0) + 500 = 500\). Thus, the fixed cost is \$500.

Step 3: Determine the Cost for a Specific Quantity of Items

To find out the cost of making 25 items, substitute \(x = 25\) into the cost function \(C(x) = 10x + 500\).

Step 4: Calculate the Cost for 25 Items

Substitute \(x = 25\) into the equation: \(C(25) = 10(25) + 500 = 250 + 500 = 750\). Hence, the cost of making 25 items is \$750.

Step 5: Understand the Domain of the Cost Function

The domain of a function refers to all the possible input values (\(x\)) that the function can accept. Since \(x\) represents the number of items, it must be a non-negative integer. The domain is therefore \(x \geq 0\).

Step 6: Understand the Range of the Cost Function

The range of a function is all possible output values. Using the condition that the maximum cost allowed is \$1500, solve \(10x + 500 \leq 1500\) to find the maximum \(x\).

Step 7: Calculate the Range of the Cost Function

Solve for \(x\): \(10x + 500 \leq 1500\) simplifies to \(10x \leq 1000\) and further to \(x \leq 100\). The range of \(C(x)\) is any value from \\(500 (when \(x = 0\)) to \\)1500 (when \(x = 100\)).

Key Concepts

Fixed Cost

The fixed cost in a cost function represents expenditure that does not vary with the level of output. In many business scenarios, this can include rent, salaries, and other overheads that remain constant regardless of production quantity.
To find the fixed cost using the given cost function, you'll need to determine the cost when no items are produced. In other words, we're seeing what cost is incurred even if production is zero.

• Set the number of items, \(x\), to 0.
• The cost function is given as \(C(x) = 10x + 500\).
• Substitute \(x = 0\). The computation is \(C(0) = 10(0) + 500 = 500\).

Therefore, the fixed cost here is \$500. This amount is paid regardless of how many items are produced.

Domain and Range

Understanding the domain and range of a function is key to fully grasping how it behaves.
For this cost function, the domain represents all potential values of \(x\), the input or number of items that can be produced.

• Since producing a negative amount of items is not feasible, the domain is all whole numbers greater than or equal to zero, \(x \geq 0\).

The range, on the other hand, is all possible output values, or costs in this context. Given the restriction of a maximum allowable cost of \\(1500, consider:

• The cost function is \(C(x) = 10x + 500\).
• Set this equal to the maximum allowable cost to find the maximum number of items: \(10x + 500 \leq 1500\).
• Solving gives \(x \leq 100\).

With \(x\) ranging from 0 to 100, the lowest cost is \\)500 (when \(x = 0\)) and the highest is \\(1500 (when \(x = 100\)). Therefore, the range of the function is all values between \\)500 to \$1500.

Linear Functions

Linear functions are mathematical expressions that model relationships with constant rates of change. They're called "linear" because when you graph them, they form a straight line.
In this exercise, the cost function \(C(x) = 10x + 500\) is linear. Here's why this matters:

• The coefficient of \(x\) (which is 10 in our function) is the slope of the line. It tells us the rate at which costs increase with each additional item produced.
• The term without an \(x\) (which is 500) is the y-intercept. It tells us the fixed cost, or the cost when zero items are produced.

Understanding these components helps predict and interpret changes in the cost as production scales. With linear equations like this, you can quickly calculate costs for different production levels without performing complex computations.