Q 4.59 - Intermediate Algebra

Question

The manufacturer of a weight training bench spends \($15$ to build each bench and sells them for \)$32$. The manufacturer also has fixed costs each month of $$25,500$.

1. Find the cost function $C$ when $x$ benches are manufactured.
2. Find the revenue function $R$ when $x$ benches 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. 

Step-by-Step Solution

Answer

Part a. The cost function is $C\left(x\right)=15x+25500$

Part b. The revenue function is $R\left(x\right)=32x$

Part c. The break-even point is shown by graphing both the Revenue and Cost functions on the same grid 

Part d. The Break-Even point is $(1500,48000)$, that means when the manufacturer sells $1500$ benches then the cost and revenue will both be equal to $$48000$

Part a Step 1. Finding cost function

The manufacturer has $$25,500$ of fixed costs no matter how many weight training benches it produces. In addition to the fixed costs, the manufacturer also spends $$15$ to produce each bench. Suppose $x$ benches are sold.

The general form of cost function is
$C\left(x\right)=(\cos t per unit)\times x+fixed \cos t$

On substituting the values we get

$C\left(x\right)=15x+25500$

Part b Step 1. Finding the Revenue Function

The manufacturer sells each weight training bench for $$32$. We get the total revenue by multiplying the revenue per unit times the number of units sold. Write the general Revenue function.

$R\left(x\right) = (selling price per unit)\times x$ 

Substitute in the revenue per unit.

$R\left(x\right)=32x$

Part c Step 1. Graph the two functions

The graph of the cost function $C\left(x\right)=15x+25500$

and revenue function $R\left(x\right)=32x$ on the same coordinate plane is given as

Part d Step 4. Find the break-even point

To find the actual value, we remember the break-even point occurs when costs equal revenue.  So

$\begin{array}{rcl}C\left(x\right)& =& R\left(x\right)\\ 15x+25500& =& 32x\\ 25500& =& 17x\\ 1500& =& x\end{array}$

So when $1500$ benches are sold, the cost equal the revenue and it is equal to

$\begin{array}{rcl}R\left(x\right)& =& 32x\\ R\left(1500\right)& =& 32\times 1500\\ R\left(1500\right)& =& 48000\end{array}$