Question

assume that the situation can be expressed as a linear cost function. find the cost function. fixed cost is $300; 70 items cost $1,700 to produce.

Explanation:

Step1: Recall linear cost function form

A linear cost function is typically of the form \( C(x) = mx + b \), where \( C(x) \) is the total cost, \( m \) is the variable cost per item, \( x \) is the number of items, and \( b \) is the fixed cost. We know the fixed cost \( b = 300 \).

Step2: Find the variable cost per item \( m \)

We know that when \( x = 70 \), \( C(70)=1700 \). Substitute \( b = 300 \), \( x = 70 \), and \( C(70)=1700 \) into the cost function:
\( 1700 = 70m + 300 \)
Subtract 300 from both sides:
\( 1700 - 300 = 70m \)
\( 1400 = 70m \)
Divide both sides by 70:
\( m=\frac{1400}{70}=20 \)

Step3: Write the cost function

Now that we have \( m = 20 \) and \( b = 300 \), the linear cost function is \( C(x)=20x + 300 \).

Answer:

\( C(x) = 20x + 300 \)