Question

A company makes pens. They sell each pen for $6. Their revenue is represented by R = 6 x. The cost to make the pens is $2 each with a one time start up cost of $6500. Their cost is represented by C = 2 x + 6500. a) Find the profit, P, (P = R - C) when the company sells 1000 pens.

Answer 1

Their revenue is given by:

$R(x)=6x$

The cost by:

$C(x)=2x+6500$

And the profit by:

$P(x)=R(x)-C(x)$

Since we need to find the profit when the company sellls 1000 pens, we need to evaluate the functions for x = 1000

$\begin{gathered} R(1000)=6(1000)=6000 \\ C(1000)=2(1000)+6500=2000+6500=8500 \\ so\colon \\ P(1000)=R(1000)-C(1000)=6000-8500=-2500 \end{gathered}$

In order to find the number of pens needed to sell to break even. we can use the following inequality:

$\begin{gathered} P(x)\ge0 \\ so\colon \\ 6x-(2x+6500)\ge0 \\ 6x-2x+6500\ge0 \\ 4x-6500\ge0 \\ 4x\ge6500 \\ x\ge\frac{6500}{4} \\ x\ge1625 \end{gathered}$