Question

A company produces very unusual CD's for which the variable cost is $14 per CD and the fixed costs are $ 40000. They will sell the CD's for $93 each. Let x be the number of CD's produced. C=$______Write tbe total revenue R as a function of the number of CD's produced.R=$_______Write the total profit P as a function of the number of CD's produced.P=$________Find the number of CD's which must be produced to break even.The number of CD's which must be produced to break even is ________

Answer 1

The cost function has a fixed part ($40,000) and a variable part, that depends on the number of CD's (14x), so we can write the cost function as:

$C(x)=40000+14x$

The revenue is equal to the unit price times the number of CD's:

$R(x)=93x$

The profit function is the difference between the revenue and the cost:

$\begin{gathered} P(x)=R(x)-C(x) \\ P(x)=93x-40000-14x \\ P(x)=(93-14)x-40000 \\ P(x)=79x-40000 \end{gathered}$

The number of CD's that must be produced and sold to breakeven happens when C(x)=R(x) or P(x)=0, so we can write:

$\begin{gathered} P(x)=0 \\ 79x-40000=0 \\ 79x=40000 \\ x=\frac{40000}{79} \\ x\approx506.32\approx507 \end{gathered}$

Answer:

C = $40000 - $14x

R = $93x

P = $79x - $40000

The breakeven number of CD's is 507 units.