Question

Assume that the below situation can be expressed as a linear cost function. Find the cost function in each case: Fixed cost:$\$100$; $50$ items cost $\$1600$ to produce.

Answer 1

Step-1: Solve for the value of the slope of linear function:

The general equation is given by,

$y=mx+c$ where, $m$ is the slope and $c$ is the $y$-intercept.

According to the question,

$y$ is the total cost of production, $x$ is the number of items and $c$ is the fixed cost

$$\begin{gathered} \therefore y=1600 \\ x=50 \\ c=100 \end{gathered}$$

Substituting the given values, we get,

$\begin{array}{rcl}1600& =& 50m+100\\ & \Rightarrow & 1600-100=50m\\ \Rightarrow 1500& =& 50m\\ & \Rightarrow & m=\frac{1500}{50}\\ & \Rightarrow & m=30\end{array}$

Step-2: Solve for the required cost function:

Let the cost function be represented as $C\left(x\right)$.

Therefore, the function will be,

$$\begin{gathered} C\left(x\right)=mx+c \\ C\left(x\right)=30x+100 \end{gathered}$$

Hence, the cost function is $C\left(x\right)=30x+100$.