Question

The manufacturer of an energy drink spends \($1.20$to make each drink and sells them for \)$2$. The manufacturer also has fixed costs each month of $$8,000$.

1. Find the cost function $C$when $x$energy drinks are manufactured.
2. Find the revenue function $R$when $x$drinks are sold.
3. Show the break-even point by graphing both the Revenue and Cost functions on the same grid.
4. Find the break-even point. Interpret what the break-even point means.

Short answer

Part(a) The cost function is $C\left(x\right)=1.2x+8000$.

Part(b) The revenue function is $R\left(x\right)=2x$.

Part(c) The break even point is given by the following graph of revenue and cost functions,

Part (d) The break-even point is $(10000,20000)$.For $10,000$drinks manufactured, the revenue and cost both are equal to $\$20,000$.

Step-by-step solution

Part(a) Step 1. Given Information.

We are given that the cost of energy drink is $\$1.20$and the selling price is $\$2$.

The manufacturer has fixed cost equal to $\$8,000$.

Part(a) Step 2. Finding the cost function.

Let $x$be the number of drink sold,

The manufacturer has fixed cost equal to $\$8,000$per month, so the cost function will be,

$C\left(x\right)=1.2x+8000$

Part(b) Step 1. Finding the revenue function R.

The manufacturer sells each drink for $\$2$, so the revenue function will be,

$R\left(x\right)=2x$

Part(c) Step 1. Graphing both the Revenue and Cost functions

The break-even point is given by the following graph,

Part(d) Step 1. Interpreting the break even point

The break even point is the solution of cost function and revenue function, so

$$\begin{gathered} 1.2x+8000=2x \\ 2x-1.2x=8000 \\ 0.8x=8000 \\ x=\frac{8000}{0.8} \\ x=10,000 \end{gathered}$$

Now, putting the value of $x$in any function, we get

$$\begin{gathered} R=2x \\ R=2\times 10000 \\ R=20,000 \\ C=1.2\left(10000\right)+8000 \\ C=12,000+8,000 \\ C=20,000 \end{gathered}$$

Hence when $10,000$energy drinks are sold then the cost and revenue both equal to $\$20,000$.