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, $C (x)$ ?

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Solution

Step-1: (a) Determine the fixed cost of the item:
The cost function is $C (x) = 10 x + 500$ .
Substitute $0$ for $x$ in the cost function.
$\begin{array}{rcl} C (0) & = & 10 \times 0 + 500 \\ = & 500 \end{array}$
Hence, the fixed cost of the item is $500$ dollars.
Step-2: (b) Determine the cost when it produces $25$ items.
Substitute $25$ for $x$ in the cost function.
$\begin{array}{rcl} C (25) & = & 10 \times 25 + 500 \\ = & 250 + 500 \\ = & 750 \end{array}$
Hence, the cost of making $25$ items is $750$ dollars.
Step-3: (c) Determine the domain and range of the cost function $C (x)$ .
Substitute $1500$ for $C (x)$ in the cost function.
$\begin{array}{rcl} 1500 & = & 10 x + 500 \\ 10 x & = & 1500 - 500 \\ = & 1000 \\ x & = & 100 \end{array}$
In order to attain a maximum cost of $$ 1500$ , the domain should be between $0$ to $100$ items and the range should be between $$ 500 to $ 1500$ .
Hence, the domain and range of the cost function is $[0, 100]$ and $[500, 1500]$ .

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