Question

you invest in a new play. the cost includes an overhead of $37,500, plus production costs of $2500 per performance. a sold - out performance brings in $3125. (in solving let x represent the number of sold - out performances.)
a. write the cost function, c.
b. write the revenue function, r.
c. determine the break - even point. describe what this means.
r(x)=3125x (type an expression using x as the variable.)
c. determine the break - even point. 6018750 (type an ordered pair. do not use commas in the individual coordinates.)
describe what this means.
a. the point where revenue and overhead cost are equal.
b. the point where the cost and overhead cost are equal.
c. the point where the cost and overhead cost are equal.
d. the point where revenue and production cost are equal.

Explanation:

Step1: Define cost function

The overhead cost is $37500$ and production cost per performance is $2500$. So the cost function $C(x)=37500 + 2500x$.

Step2: Define revenue function

A sold - out performance brings in $3125$, so the revenue function $R(x)=3125x$.

Step3: Find break - even point

Set $C(x)=R(x)$. So $37500 + 2500x=3125x$. Subtract $2500x$ from both sides: $37500=3125x - 2500x$, which simplifies to $37500 = 625x$. Solve for $x$: $x=\frac{37500}{625}=60$. Substitute $x = 60$ into $R(x)$ (or $C(x)$), $R(60)=3125\times60 = 187500$. The break - even point is $(60,187500)$.

Step4: Interpret break - even point

The break - even point is the point where cost and revenue are equal.

Answer:

a. $C(x)=37500 + 2500x$
b. $R(x)=3125x$
c. $(60187500)$
D. The point where cost and revenue are equal.