Question

The manufacturer of a water bottle spends \($5$to build each bottle and sells them for \)$10$. The manufacturer also has fixed costs each month of $$6500$.

1. Find the cost function $C$when $x$bottles are manufactured.
2. Find the revenue function $R$when $x$bottles 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 $C\left(x\right)=5x+6500$.

Part(b) The revenue function is $R\left(x\right)=10x$.

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

Part(d) The break-even point is $(1300,13000)$. For $1,300$water bottles the cost and revenue both are equal to $\$13,000$.

Step-by-step solution

Part(a) Step 1. Given Information.

We are given that the cost of water bottle is $\$5$and the selling price is $\$10$.

The manufacturer has fixed cost equal to $\$6,500$.

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

Let $x$be the number of drink sold,

The manufacturer has fixed cost equal to$\$6500$ per month, so the cost function will be,

$C\left(x\right)=5x+6500$

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

The manufacturer sells each drink for $\$10$, so the revenue function will be,

$R\left(x\right)=10x$

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

$$\begin{gathered} 5x+6500=10x \\ 10x-5x=6500 \\ 5x=6500 \\ x=\frac{6500}{5} \\ x=1300 \end{gathered}$$

Now, putting the value of $x$in revenue and cost function, we get

$$\begin{gathered} R\left(1300\right)=10\times 1300 \\ R\left(1300\right)=13,000 \\ C\left(1300\right)=5\times 1300+6500 \\ C\left(1300\right)=6500+6500 \\ C\left(1300\right)=13,000 \end{gathered}$$

Hence, when $1,300$water bottles are manufactured then the cost and revenue both equal to $\$13,000$.