A cost function, [tex]C(x)[/tex], is given below. Find the average cost per item when the required number of items are produced.
[tex]C(x)=20 x+1500,100[/tex] items
What is the average cost per item?
Answer (1)
Calculate the total cost by substituting x = 100 into the cost function: C ( 100 ) = 20 ( 100 ) + 1500 = 3500 .
Divide the total cost by the number of items to find the average cost per item: 100 3500 = 35 .
The average cost per item is $35.
The final answer is 35 .
Explanation
Understanding the Problem We are given the cost function C ( x ) = 20 x + 1500 , where x is the number of items produced. We need to find the average cost per item when 100 items are produced.
Calculating the Total Cost First, we need to calculate the total cost of producing 100 items. We substitute x = 100 into the cost function: C ( 100 ) = 20 ( 100 ) + 1500
Total Cost Calculation Now, we perform the calculation: C ( 100 ) = 2000 + 1500 = 3500 So, the total cost of producing 100 items is $3500.
Calculating the Average Cost Next, we need to find the average cost per item. To do this, we divide the total cost by the number of items produced: Average Cost = 100 C ( 100 ) = 100 3500
Average Cost Calculation Now, we perform the division: Average Cost = 100 3500 = 35 Therefore, the average cost per item is $35.
Final Answer The average cost per item when 100 items are produced is 35 .
Examples
Imagine you are running a small business that produces handmade crafts. Your cost function helps you understand how much it costs to produce each item, including materials and other expenses. By calculating the average cost per item, you can determine a suitable selling price to ensure you cover your costs and make a profit. For example, if your cost function is C ( x ) = 5 x + 500 , where x is the number of crafts, and you produce 20 crafts, the total cost is $C(20) = 5(20) + 500 = $600. The average cost per craft is $600 / 20 = $30, which you can use to set your selling price.
