Q.A manufacturer's total cost is a linear function of output. When output is 10 units, total cost is ₹800; when output is 20 units, total cost is ₹1300. Find the cost function $C(x)$ and state the fixed cost.

A linear cost function can be determined from any two known (output, cost) pairs, exactly like finding the equation of a straight line through two points.

Since total cost is a linear function of output, write $C(x) = mx+c$, where $m$ is the variable cost per unit and $c$ is the fixed cost. We are given two points on this line: $(10, 800)$ and $(20, 1300)$.

Finding $m$ (the slope):

So the variable cost per unit is $₹50$, and $C(x) = 50x + c$ so far.

Finding $c$ (the fixed cost): substitute either known point into $C(x)=50x+c$. Using $(10,800)$:

So the cost function is

Dual check using the other point, $(20, 1300)$:

$C(20) = 50(20)+300 = 1000+300 = 1300$ …

A common slip is finding the slope correctly but then substituting the wrong pair of coordinates when solving for $c$ (e.g. mixing up which value is $x$ and which is $C(x)$) — it always helps to re-check the …