Question

assume that the situation can be expressed as a linear cost function. find the cost function.
fixed cost is $200; 40 items cost $1,000 to produce.

Explanation:

Step 1: Recall the linear cost function formula

A linear cost function is generally in the form \( C(x) = mx + b \), where \( C(x) \) is the total cost, \( m \) is the marginal cost (cost per item), \( x \) is the number of items, and \( b \) is the fixed cost. We know the fixed cost \( b = 200 \).

Step 2: Find the marginal cost \( m \)

We know that when \( x = 40 \), \( C(40)=1000 \). Substitute \( b = 200 \), \( x = 40 \), and \( C(40) = 1000 \) into the cost function:
\[
1000 = m(40)+200
\]
Subtract 200 from both sides:
\[
1000 - 200=40m
\]
\[
800 = 40m
\]
Divide both sides by 40:
\[
m=\frac{800}{40}=20
\]

Step 3: Write the cost function

Now that we have \( m = 20 \) and \( b = 200 \), substitute these into the linear cost function \( C(x)=mx + b \):
\[
C(x)=20x + 200
\]

Answer:

The cost function is \( C(x) = 20x + 200 \)